CryptographyNetSecurity-2008

Cryptography and Network Security 1

CS549:Cryptography and Network Security

© by Xiang-Yang Li

Department of Computer Science, IIT


Notice©

This lecture note (Cryptography and Network Security) is prepared by Xiang-Yang Li. This lecture note has benefited from numerous textbooks and online materials. Especially the “Cryptography and Network Security” 2nd edition by William Stallings and the “Cryptography: Theory and Practice” by Douglas Stinson.

You may not modify, publish, or sell, reproduce, create derivative works from, distribute, perform, display, or in any way exploit any of the content, in whole or in part, except as otherwise expressly permitted by the author.

The author has used his best efforts in preparing this lecture note. The author makes no warranty of any kind, expressed or implied, with regard to the programs, protocols contained in this lecture note. The author shall not be liable in any event for incidental or consequential damages in connection with, or arising out of, the furnishing, performance, or use of these.


About Instructor

Associate Professor IIT PhD/MS UIUC 1997-2000 BS, BE Tsinghua University

Research Interests: Algorithm design and analysis Wireless networks Game theory Computational geometry

Contact Information Phone 312-567-5207 Email: [email protected]


Office and Office hours

Office SB 237D

Office hours Monday 3:10PM – 4:10PM. Wednesday 3:10PM– 4:10PM. Or by contact: email [email protected], phone 312 567 5207


About This Course Textbook

Cryptography: Theory and Practice

by Douglas R. Stinson CRC press

Cryptography and Network Security: Principles and Practice; By William Stallings Prentice Hall

Handbook of Applied Cryptography by Alfred J. Menezes, Paul C. van Oorschot and Scott A. Vanstone, CRC Press I have electronic version!

http://www.cacr.math.uwaterloo.ca/~dstinson/gifs/cover.gif

http://www.amazon.com/gp/product/images/0138690170/ref=dp_primary-product-display_0/104-8278000-3955142?%5Fencoding=UTF8&n=283155&s=books

http://www.cacr.math.uwaterloo.ca/hac/about/cover.gif


Grading and Others

Grading Homework 30% Mid Term 25% Project 20% (select your own topic),

15 pages report Final exam 25% (closed book)

Policy Do it yourself Can use library, Internet and so on, but you have to cite the

sources when you use this information


Homeworks

Do it independently No discussion No copy Can use reference books

Staple your solution Write your name also,

For report, you could discuss with

classmates then write your own report (about 10 pages for the topic you selected)

For project (presentation and programming) You SHOULD collaborate

with your group member and you SHOULD make enough contributions to get credit

HW1 (Due 2/14/08) HW2 (Due 3/14/08) HW3 (Due 4/11/08)

Report (Due 05/05/08)

Type your solution!

And print it then submit


Topics

Introduction Number Theory Traditional Methods: secret key system Modern Methods: Public Key System Digital Signature and others Internet Security: DoS, DDoS Other topics:

secret sharing, zero-knowledge proof, bit commitment, oblivious transfer,…


Organization

Chapters Introduction Number Theory Conventional Encryption Block Ciphers Public Key System Key Management Hash Function and Digital Signature Identification Secret Sharing Pseudo-random number Generation Email Security Internet Security Others


Cryptography and Network Security

Introduction

Xiang-Yang Li


Introduction

The art of war teaches us not on the likelihood of the enemy’s not coming, but on our own readiness to receive him; not on the chance of his not attacking, but rather on the fact that we have made our position unassailable.

--The art of War, Sun Tzu


Criteria for Desirable Cryptosystems Confidence in Security established

Hard or intractable problems? Practical Efficiency

Space, time and so on Explicitness

About its environment assumptions, security service offered, special cases in math assumptions,

Protection tuned to application needs No less, no more

Openness


Most important

Security first

Efficiency, resource utilization, and security tradeoffs This is especially the case for resource constrained

networks such as wireless sensor networks Limited power supply (thus limited

communication, and computation), limited storage space


Cryptography

Cryptography (from Greek kryptós, "hidden", and gráphein, "to write") is, traditionally, the study of means of converting information from its normal, comprehensible form into an incomprehensible format, rendering it unreadable without secret knowledge — the art of encryption.

Past: Cryptography helped ensure secrecy in important communications, such as those of spies, military leaders, and diplomats.

In recent decades, cryptography has expanded its remit in two ways mechanisms for more than just keeping secrets: schemes like digital

signatures and digital cash, for example. in widespread use by many civilians, and users are not aware of it.


Crypto-graphy, -analysis, -logy

The study of how to circumvent the use of cryptography is called cryptanalysis, or codebreaking.

Cryptography and cryptanalysis are sometimes grouped together under the umbrella term cryptology, encompassing the entire subject.

In practice, "cryptography" is also often used to refer to the field as a whole; crypto is an informal abbreviation.

Cryptography is an interdisciplinary subject, linguistics Mathematics: number theory, information theory, computational

complexity, statistics and combinatorics engineering


Close, but different fields

Steganography the study of hiding the very existence of a message, and not

necessarily the contents of the message itself (for example, microdots, or invisible ink)

http://en.wikipedia.org/wiki/Steganography Traffic analysis

which is the analysis of patterns of communication in order to learn secret information The messages could be encrypted

http://en.wikipedia.org/wiki/Traffic_analysis


Stenography Example

Last 2 bits


Tools for Stenography

http://www.jjtc.com/Steganography/toolmatrix.htm


Network Security Model

Trusted Third Party

principal principal

Security transformation

Security transformation

attacker


Attacks, Services and Mechanisms

Security Attacks Action compromises the information security Could be passive or active attacks

Security Services Actions that can prevent, detect such attacks. Such as authentication, identification, encryption, signature, secret

sharing and so on.

Security mechanism The ways to provide such services Detect, prevent and recover from a security attack


Attacks

Passive attacks Interception

Release of message contents Traffic analysis

Active attacks Interruption, modification, fabrication

Masquerade Replay Modification Denial of service


Information Transferring


Attack: Interruption

Cut wire lines,Jam wireless

signals,Drop packets,


Attack: Interception

Wiring, eavesdrop


Attack: Modification

interceptReplaced

info


Attack: Fabrication

Also called impersonation


Attacks, Services and Mechanisms

Security Attacks Action compromises the information security Could be passive or active attacks

Security Services Actions that can prevent, detect such attacks. Such as authentication, identification, encryption, signature, secret

sharing and so on.

Security mechanism The ways to provide such services Detect, prevent and recover from a security attack


Important Services of Security Confidentiality, also known as secrecy:

only an authorized recipient should be able to extract the contents of the message from its encrypted form. Otherwise, it should not be possible to obtain any significant information about the message contents.

Integrity: the recipient should be able to determine if the message has

been altered during transmission.

Authentication: the recipient should be able to identify the sender, and verify

that the purported sender actually did send the message.

Non-repudiation: the sender should not be able to deny sending the message.


Secure Communication

protecting data locally only solves a minor part of the problem. The major challenge that is introduced by the Web Service security requirements is to secure data transport between the different components. Combining mechanisms at different levels of the Web Services protocol stack can help secure data transport (see figure next page).





The combined protocol HTTP/TLS or SSL is often referred to as HTTPS (see figure). SSL was originally developed by Netscape for secure communication on the Internet, and was built into their browsers. SSL version 3 was then adopted by IETF and standardized as the Transport Layer Security (TLS) protocol.

Use of Public Key Infrastructure (PKI) for session key exchange during the handshake phase of TLS has been quite successful in enabling Web commerce in recent years.

TLS also has some known vulnerabilities: it is susceptible to man-in-the-middle attacks and denial-of-service attacks.


SOAP security

SOAP (Simple Object Access Protocol) is designed to pass through firewalls as HTTP. This is disquieting from a security point of view. Today, the only way we can recognize a SOAP message is by parsing XML at the firewall. The SOAP protocol makes no distinction between reads and writes on a method level, making it impossible to filter away potentially dangerous writes. This means that a method either needs to be fully trusted or not trusted at all.

The SOAP specification does not address security issues directly, but allows for them to be implemented as extensions. As an example, the extension SOAP-DSIG defines the syntax and

processing rules for digitally signing SOAP messages and validating signatures. Digital signatures in SOAP messages provide integrity and non-repudiation mechanisms.


PKI

PKI key management provides a sophisticated framework for securely exchanging and managing keys. The two main technological features, which a PKI can provide to Web Services, are: Encryption of messages: by using the public key of the recipient Digital signatures: non-repudiation mechanisms provided by PKI and

defined in SOAP standards may provide Web Services applications with legal protection mechanisms

Note that the features provided by PKI address the same basic needs as those that are recognized by the standardization organizations as being important in a Web Services context.

In Web Services, PKI mainly intervenes at two levels: At the SOAP level (non-repudiation, integrity) At the HTTPS level (TLS session negotiation, eventually assuring

authentication, integrity and privacy)


Some basic Concepts


Cryptography

Cryptography is the study of Secret (crypto-) writing (-graphy)

Concerned with developing algorithms: Conceal the context of some message from all except

the sender and recipient (privacy or secrecy), and/or Verify the correctness of a message to the recipient

(authentication) Form the basis of many technological solutions to

computer and communications security problems


Basic Concepts

Cryptography encompassing the principles and methods of transforming

an intelligible message into one that is unintelligible, and then retransforming that message back to its original form

Plaintext The original intelligible message

Ciphertext The transformed message

Message Is treated as a non-negative integer hereafter


Basic Concepts

Cipher An algorithm for transforming an intelligible message

into unintelligible by transposition and/or substitution, or some other techniques

Keys Some critical information used by the cipher, known

only to the sender and/or receiver Encipher (encode)

The process of converting plaintext to ciphertext Decipher (decode)

The process of converting ciphertext back into plaintext


Basic Concepts

cipher an algorithm for encryption and decryption. The exact

operation of ciphers is normally controlled by a key — some secret piece of information that customizes how the ciphertext is produced

Protocols specify the details of how ciphers (and other cryptographic

primitives) are to be used to achieve specific tasks. A suite of protocols, ciphers, key management, user-

prescribed actions implemented together as a system constitute a cryptosystem;

this is what an end-user interacts with, e.g. PGP


Encryption and Decryption

Plaintext ciphertext

Encipher C = E(K1)(P)

Decipher P = D(K2)(C)

K1, K2: from keyspace

These two keys could be different; could be difficult to get one from the other


What is Security?

Two fundamentally different securities Unconditional security

No matter how much computer power is available, the cipher cannot be broken

Using Shannon’s information theory The entropy of the message I(M) is same as the entropy of the

message I(M|C) when known the ciphertext (and possible more) Computational security

Given limited computing resources (e.g time needed for calculations is greater than age of universe), the cipher cannot be broken

What do we mean “broken”? Proved by some complexity equivalence approach



Elementary Number TheoryXiang-Yang Li


Number theory

Elementary number theory Main topic of this course divisibility, the Euclidean algorithm to compute

greatest common divisors, factorization Fermat's little theorem and Euler's theorem, the Chinese

remainder theorem and Euler's φ function are investigated;

Analytic number theory Algebraic number theory Geometric number theory Computational number theory


Introduction to Number Theory

Divisors b|a if a=mb for an integer m b|a and c|b then c|a b|g and b|h then b|(mg+nh) for any integer m,n

Prime number P has only positive divisors 1 and p

Relatively prime numbers No common divisors for p and q except 1


Prime numbers

Upto 200 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97

101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199

Largest known so far (till 2008, Jan 22) 232582657-1 with 9808358 digits (found 2006 using proof code G9) When 2n-1is prime it is said to be a Mersenne prime (a French monk

1588-1648, conjecture 1644). Clearly n must be odd. How many prime numbers are there?

Infinity ---- Euclid gave simple proof Proof by contradiction They were also irregularly placed (arbitrary gap)

How many in the range [0,n] -- Theta( n / log n) Approximately, the nth prime n log n

How many primes with d bits approximately? ~ Theta(2d/d)


Determining Primes?

How to determine if a given number n is prime? Deterministic Brute force testing

Testing whether a number a | n, for a in certain range

Random testing A prime number should satisfy some properties If a number x does NOT have any of such

properties, then this x is NOT a primeOtherwise, it may be a prime number

Properties: for any number a, a does not divide x, • More properties will be studied and used to design

efficient methods


Greatest Common Divisor (GCD) Greatest common divisor gcd(a,b)

The largest number that divides both a and b

Euclid's algorithm Find the GCD of two numbers a and b, a<b

Use fact if a and b have divisor d so does a-b, a-2b …

d m a n b

d a b

d a b

d a b

d a q b

2

3


Cont.

GCD (a,b) is given by: let g0=b

g1=a

gi+1 = gi-1 mod gi

when gi =0 then gcd(a,b) = gi-1

The algorithm terminates in O(log b) rounds Why? Every round, the total number of bits of a and b is decreased by at

least one

What is a more precise complexity bound?


Properties

For any two integers a and b Exist integers m and n: gcd(a,b) =ma+bn Example:

a=2, b=3; we choose m=-1, n=1 so –2+3=1 a=6, b=11; we choose m=2, n=-1 so 2*6-11=1

Simple proof?

Integer n can be factored as n=p1

a1 p2

a2 p3

a3…. pn

an where pi is prime number


Extended Euclidean Algorithm

input are two integers a and b, computes their greatest common divisor (gcd) as well as integers x and y such that ax + by = gcd(a, b).

It later can also be used to compute the inverse of an integer

a n 1 m od


Proof

Assume we compute gcd(x0,y0), x0>y0

Let Xi=(xi,yi); 0xi-qi+1yi+1<|yi|

Then Xi=MiXi-1, where Mi=(0,1; 1,-qi)

Assume the gcd algorithm terminates in n steps We have MnMn-1

…M1X0=(gcd(x0,y0), 0)T

Assume MnMn-1…M1=( )

Then ax0+by0=gcd(x0,y0)

The above algorithm is to keep track of a,b,c,d, and xi,yi values.

a b

c d


Modular Arithmetic

Congruence a b mod n says when divided by n that a and b have

the same remainder It defines a relationship between all integers

a a a b then b a a b, b c then a c


Cont.

addition (a+b) mod n (a mod n) + (b mod n)

subtraction a-b mod n a+(-b) mod n

multiplication a b mod n derived from repeated addition Possible: a*b 0 where neither a, b 0 mod n

Example: 2*3 =0 mod 6


Addition and Multiplication

Integers modulo n with addition and multiplication form a commutative ring with the laws of Associativity

(a+b)+c a+(b+c) mod n Commutativity

a+b b+a mod n Distributivity

(a+b)*c (a*c)+(b*c) mod n


Cont.

Division b/a mod n multiplied by inverse of a: b/a = b*a-1 mod n a-1*a 1 mod n 3-1 7 mod 10 because 3*7 1 mod 10 Inverse does not always exist!

Only when gcd(a,n)=1


Euclid's Extended GCD Routine If (a,n)=1 then the inverse always exists Can extend Euclid's algorithm to find

inverse by keeping track of gi = ui.n + vi.a

Extended Euclid's (or binary GCD) algorithm to find inverse of a number a mod n (where (a,n)=1) is:


Inverse

Inverse(a,n) is given by: X=(x1,x2,x3)=(1,0,n); Y=(y1,y2,y3)=(0,1,a) If y3=0 return x3=gcd(a,n); no inverse If y3=1 return y3=gcd(a,n); y2=a-1 mod n Q=[x3/y3] T=X-Q*Y X=Y; Y=T Goto 2nd step


When inverse exists

If gcd(a,n)=1 inverse exists We can find x, y such that ax+ny=1 Then x= a-1 mod n

If inverse exists gcd(a,n)=1 Let x be the inverse of a, i.e., ax=1 mod n Then x a=1+q n for some integer q Let gcd(a,n)=d. Then d | (x a-q n ) Obviously d=1 since x a-q n =1


Galois Field

If n is constrained to be a prime number p then this forms a Galois field modulo p denoted GF(p) and all the normal laws associated with integer arithmetic work

Exponentiation b = ae mod p

Discrete Logarithms find x where ax = b mod p


Relative primes

Two numbers a and n are relative primes if gcd(a,n)=1

Consider all integers 0<a <n How many are relative prime to n? Equivalently, how many a such that a-1 mod n exists

Typically Zn={0,1,2,….,n-1} : all integers 0<= a < n

Zn*={a| 0<= a < n, gcd(a,n)=1}

All integers in Zn that are co-prime with n Also called reduced residue set mod n


Euler Totient Function

If consider arithmetic modulo n, then a reduced set of residues is a subset of the complete set of residues modulo n which are relatively prime to n eg for n=10, the complete set of residues is {0,1,2,3,4,5,6,7,8,9} the reduced set of residues is {1,3,7,9}

The number of elements in the reduced set of residues is called the Euler Totient function (n)


cont

Compute (n) If factoring of n is known

(n)=n (1-1/pi) where pi is its prime factor

Otherwise It is expensive! But not proved yet

computing (n) when knowing fact n =pq but not the number p and q Conjectured to be a hard question But not proved yet. Equivalent to find p and q


cont

Equivalency: finding p,q computing (n)

Proof If we found p and q, then (n)=(p-1)(q-1) if we found (n), then solve p, q from equations

n p q

n p q

( ) ( )( )1 1


Euler's Theorem

Let gcd(a,n)=1 then a(n) mod n = 1

Proof: consider all reduced residues xi in

Zn*={x| 0<= x < n, gcd(x,n)=1}

Then axi,1<=i <= (n) also form reduced residues set

Using axi = xi mod n Using Zn

* and aZn* are same sets!

We have a(n) xi = xi mod n

Thus, a(n) =1 mod n Using the fact that xi has inverse


Fermat's Little Theorem

Let p be a prime and gcd(a,p)=1 then ap-1 mod p = 1 Proof: similar to the proof of Euler’s theorem But consider all integers in Zp

Generally, for any prime number p ap mod p = a (true for any number a)

Generally, for any number n=pq a(n) mod n = a (true for any number a)

Need to prove for the case gcd(a,n)>1

Do it yourself


Efficient computing of exponential Compute ab mod n efficiently when b, n

large? Example: compute a1024 mod 21024 +1 Simple approach: repetitively time a 1024 times? Efficient computation:

Write number b in binary format as xkxk-1xk-2….x2x1x0

Let t1=a mod n. Then compute ti+1= ti * ti mod n for i<k

Then

a n a n

a n

t n

b x x x x x x

x

i k

ix

i k

k k k

ii

i

m od m od

[ ] m od

m od

....

( )

1 2 2 1 0

2

0

0

Time complexity?


Chinese Remainder Theorem

By Qin Jiushao Let m1,m2,….mk be pair-wise relative prime numbers Assume integer x= ai mod mi for 1<= I <= k Then x= ai ei mod M

Where M= mi; Mi=M/ mi

ei= Mi * (Mi-1 mod mi)

Proof For each i, the integers mi and M/mi are coprime, and using the extended Euclidean

algorithm we can find integers r and s such that r mi + s M/mi = 1. If we set ei = s M/mi, then we have

ei =1 mod mi and ei =1 mod mj for j<>i.


General CRT

Sometimes, the simultaneous congruences can be solved even if the mi's are not pairwise coprime. a solution x exists if and only if ai ≡ aj (mod gcd(ni, nj))

for all i and j. All solutions x are congruent modulo the least common

multiple of the ni.

Methods: successive substitution


Example

consider the simultaneous congruences x ≡ 3 (mod 4) x ≡ 5 (mod 6)

Can be transformed to x ≡ 3 (mod 4) x ≡ 5 (mod 2) x ≡ 1 (mod 2) x ≡ 5 (mod 3)

Then transformed to x ≡ 3 (mod 4) x ≡ 2 (mod 3)

Using CRT X=11 (mod 12)


Primality Testing

To check if exists integer a such that a|n Primary school method

Test a=2,3,4,5,6,….,n-1 Test a=2,3,4,5,…, n0.5

Test a=2,3,5,7,11,…., p, where prime number p<=n0.5

Two slow! Check almost n numbers Check n0.5 numbers At least around (n/ln n)0.5 numbers need be checked

Example Number n~21024, then (n/ln n)0.5~(21024 /1024) 0.5 ~ 2507

Assume 230 numbers per second, takes about 2507-30*16 = 227 days Any improvement?


Classification of Testing Primes

The Quick Tests for Small Numbers and Probable Primes Finding Very Small Primes --- trivial division Fermat, Probable-Primality and Pseudoprimes Strong Probable-Primality and a Practical Test

The Classical Tests N-1 Tests (and Pepin's Test for Fermats) N+1 Tests (and the Lucas-Lehmer Test for Mersennes) A Combined Test -- and more

The General Purpose Tests Neoclassical Tests, especially APR and APR-CL Using Elliptic Curves, especially the ECPP Test A Polynomial Time Algorithm


Fermat Little Theorem Based

Fermat's theorem gives us a powerful test for compositeness: Given n > 1, choose a > 1 and calculate an-1 modulo n

(there is a very easy way to do quickly by repeated squaring)

If the result is not one modulo n, then n is composite. If it is one modulo n, then n might be prime so n is

called a weak probable prime base a (or just an a-PRP).

Some early articles call all numbers satisfying this test pseudoprimes, but now the term pseudoprime is properly reserved for composite probable-primes.


Carmichael number

There may be relatively few pseudoprimes, but there are still infinitely many of them for every base a>1, so we need a tougher test.

One way to make this test more accurate is to use multiple bases (check base 2, then 3, then 5,...). But still we run into an interesting obstacle called the Carmichael numbers. The composite integer n is a Carmichael number if

an-1=1 (mod n) for every integer a relatively prime to n.


Strong probable-primality and a practical test A better way to make the Fermat test more

accurate is to realize that if an odd number n is prime, then the number 1 has just two square roots modulo n: 1 and -1. So the square root of an-1, a(n-1)/2 (since n will be odd), is either 1 or

-1.

Algorithm Write n-1 = 2sd where d is odd and s is non-negative: n is a strong

probable-prime base a (an a-SPRP) if either ad = 1 (mod n) or (ad)2r = -1 (mod n) for some non-negative r less than s.

It has been proven ([Monier80] and [Rabin80]) that the strong probable primality test is wrong no more than 1/4th of the time (3 out of 4 numbers which pass it will be prime).

http://primes.utm.edu/references/refs.cgi/Monier80

http://primes.utm.edu/references/refs.cgi/Rabin80


Simple Fact

Equation x21 mod p has only solutions 1,-1 If p is prime number Simple proof: (x+1)(x-1) 0 mod p

So if we find another solution, then p can not be prime number! Miller and Rabin 1975,1980

Randomly chosen integer a If a21 mod p then p is not prime number

Integer a is called the witness Otherwise p maybe, or maybe not a prime number


Witness Algorithm

Witness(a,n) Let bkbk-1…b1b0 be the binary code of n-1

Let d=1 For i=k downto 0 x=d; d=d*d mod n If d=1 and x1, and x n-1 return TRUE If bi=1 then d=d*a mod n

Endfor If d 1 then return TRUE Return FALSE


Facts

Analyze the result of witness If returns TRUE, then n is not prime number

Find other solutions for x21 mod n Otherwise, n maybe prime number

Given odd n and random a Witness fails with probability less than 0.5

Run witness algorithm s times If one time, it is TRUE

Then n is not prime number Otherwise, Pr(n is prime)>1-2-s


Randomized Methods

Las Vegas Method Always produces correct results Runs in expected polynomial time

Monte Carlo Method Runs in polynomial time May produce incorrect results with bounded probability No-Biased Monte Carlo Method

Answer yes is always correct, but the answer no may be wrong

Yes-biased Monte Carlo Method Answer no is always correct, but the answer yes may

be wrong


Witness Algorithm

Witness Algorithm is based on Monte Carlo Method It actually test compositeness, not primality

When it reports yes, the number is always composite

When it reports no, input may be composite, prime

Probability Result Pr(input=composite | ans=composite)= 1 Pr(ans=no | input=composite)<1/2 Pr(input=composite | ans=no) 1/4


Time Complexity

Each round of witness cost O(log n) Unit: integer multiplication and modular arithmetic

So the primality testing cost O(s log n) The confidence is 1-2-s if report prime The confidence is 1 if report non-prime

Miller's Test [Miller76]: If the extended Riemann hypothesis is true, then if n is an a-SPRP for all integers a with 1 < a < 2(log n)2, then n is prime.

http://primes.utm.edu/references/refs.cgi/Miller76


More on proving primes (N-1 test Theorem 1: Let n > 1. If for every

prime factor q of n-1 there is an integer a such that an-1 = 1 (mod n), and a(n-1)/q is not 1 (mod n);

then n is prime.


N-1 test

Theorem 2: Suppose n-1 = FR, where F>R, gcd(F,R) is one and the factorization of F is known. If for every prime factor q of F there is an integer a>1 such that an-1 = 1 (mod n), and gcd(a(n-1)/q-1,n) = 1;

then n is prime.


N+1 test

Lucas-Lehmer Test (1930): Let n be an odd prime. The Mersenne number M(n) = 2n-1 is prime if and only if S(n-2) = 0 (mod M(n)) where

S(0) = 4 and S(k+1) = S(k)2-2.


ECPP method

What is the next big leap in primality proving? To switch from Galois groups to some other, perhaps easier to work with groups--in this case the points on Elliptic Curves modulo n. An Elliptic curve is a curve of genus one, that is a curve that can

be written in the form E(a,b) : y2 = x3 + ax + b (with 4a3 + 27b2 not zero) http://www.lix.polytechnique.fr/~morain/Prgms/ecpp.english.html

for implementation

Heuristically, the best version of ECPP is O((log n)4+eps) for some eps>0

http://www.lix.polytechnique.fr/~morain/Prgms/ecpp.english.html


Deterministic Poly-Time Method In 2002 Agrawal, Kayal and Saxena

found a relatively simple deterministic algorithm which relies on no unproved assumptions. There has been a long list of research efforts devoted to

find deterministic polynomial time methods for testing primes


Basics

Theorem: Suppose that a and p are relatively prime integers with p > 1. p is prime if and only if (x-a)p = (xp-a) (mod p)

Proof. If p is prime, then p divides the binomial coefficients pCr for r =

1, 2, ... p-1. This shows that (x-a)p = (xp-ap) (mod p), and the equation above follows via Fermat's Little Theorem.

On the other hand, if p > 1 is composite, then it has a prime divisor q. Let qk be the greatest power of q that divides p. Then qk does not divide pCq and is relatively prime to ap-q, so the coefficient of the term xq on the left of the equation in the theorem is not zero, but it is on the right.


AKS method

Input: Integer n > 1

if (n is has the form ab with b > 1) then output COMPOSITE

r := 2while (r < n) { if (gcd(n,r) is not 1) then output COMPOSITE if (r is prime greater than 2) then { let q be the largest factor of r-1 if (q > 4sqrt(r)log n) and (n(r-1)/q is not 1 (mod r)) then break } r := r+1}

for a = 1 to 2sqrt(r)log n { if ( (x-a)n is not (xn-a) (mod xr-1,n) ) then output COMPOSITE}

output PRIME;


Time Complexity

they proved would run in at most O((log n)12f(log log n)) time where f is a polynomial

AKS also showed that if Sophie Germain primes have the expected distribution [HL23] (and they certainly should!), then the exponent 12 in the time estimate can be reduced to 6, bringing it much closer to the (probabilistic) ECPP method. But of course when actually finding primes it is the unlisted

constants1 that make all of the difference! We will have to wait for efficient implementations of this algorithm (and hopefully clever restatements of the painful for loop) to see how it compares to the others for integers of a few thousand digits. Until then, at least we have learned that there is a polynomial-time algorithm for all integers that both is deterministic and relies on no unproved conjectures!

http://primes.utm.edu/references/refs.cgi/HL23


Primitive Root

Order of integer ordn(a) The order of a modulo n is the smallest positive k such

that ak1 mod n

Primitive Root Integer a is a primitive root of n if the order of a

modulo n is (n) Not all integers have primitive root

Example n=pq for primes p and q Prime p has (p-1) primitive roots


cont

When primitive root exists Number n in format of p, 2p, pk, 2pk for some integer k

and prime number p Otherwise the primitive root does not exist

Find a PR for p such that Let a=2, i=1 If i>k, a is a PR, otherwise go to step 3 If let i=i+1 and go to step 2;

otherwise let i=1, and a=a+1 and repeat this step 3.

p q qak

a k 1 11 . . . .

a pp q i( ) / m od 1 1


Some “hard” questions

Some questions that are assumed to be hard, will be used as bases for cryptography Integer factorization

Given n, find all its prime factors Discrete logarithm

Given g, y, and p, find x such that gxy mod p Square root

Given b, find x such that x2b mod n. Here n is not a prime number


Integer Factorization write an integer as product of prime numbers.

For example, given the number 45, the prime factorization would be 32·5. The factorization is always unique, according to the fundamental theorem of

arithmetic Given two large prime numbers, it is easy to multiply them. However, given

their product, it appears to be difficult to find the factors. This is relevant for many modern systems in cryptography. If a fast method

were found for solving the integer factorization problem, then several important cryptographic systems would be broken.

Although fast factoring is one way to break these systems, there may be other ways to break them that don't involve factoring. So it is possible that the integer factorization problem is truly hard, yet these systems can still be broken quickly.

A rare exception is the BBS generator. It has been proved to be exactly as hard as integer factorization: if you can break the generator in polynomial time then you can factorize integers in polynomial time, and vice versa


Current state of the art

If a large, n-bit number is the product of two primes that are roughly the same size, no polynomial time factoring algorithm is known the best known algorithms are sub-exponential, but

super-polynomial: asymptotic running time by the general number field sieve (GNFS) algorithm, is

Polynomial methods known for quantum computer!


Sub-exponential

There are published algorithms that are faster than O((1+ε)b) for all positive ε, i.e., sub-exponential, where b is the number of bits of the input

http://en.wikipedia.org/wiki/Big_O_notation


Factoring algorithms

Special purpose its running time depends on the properties of unknown factors:

size, special form, etc. Examples

Trial division, Pollard's rho algorithm, Pollard's p-1 algorithm, Lenstra elliptic curve factorization, Congruence of squares, Special number field sieve

General purpose running time depends solely on the size of the integer to be

factored. This is the type of algorithm used to factor RSA numbers. Most general-purpose algorithms are based on the congruence of squares method.

Examples: Quadratic sieve, General number field sieve


Factorization for Quantum Computers For an ordinary computer, general number field

sieve (GNFS) is the best published algorithm for large n (more than about 100 digits).

For a quantum computer, however, Peter Shor discovered an algorithm in 1994 that solves it in polynomial time. This will have significant implications for cryptography if a large quantum computer is ever built.

Shor's algorithm takes only O(b3) time and O(b) space on b-bit number inputs.

In 2001, the first 7-qubit quantum computer became the first to run Shor's algorithm. It factored the number 15.


List of Algorithms

Special-purpose A special-purpose factoring algorithm's running time depends on the

properties of its unknown factors: size, special form, etc. Exactly what the running time depends on varies between algorithms.

Trial division Pollard's rho algorithm Algebraic-group factorisation algorithms amongst which are Pollard's p − 1 algorithm,

Williams' p+1 algorithm and Lenstra elliptic curve factorization Fermat's factorization method Special number field sieve

General-purpose A general-purpose factoring algorithm's running time depends solely

on the size of the integer to be factored. This is the type of algorithm used to factor RSA numbers. Most general-purpose factoring algorithms are based on the congruence of squares method.

Dixon's algorithm Continued fraction factorization (CFRAC) Quadratic sieve General number field sieve Shanks' square forms factorization (SQUFOF)


Discrete Logarithms

Y gx mod p Given y, g, and p, compute x as logg(y) Time complexity O(e(ln p)1/3(ln ln p)2/3)

Best known until now In other words, if p is large, then it is very hard to solve the discrete logarithm problem

Several protocols are based on this ElGamal discrete log cryptosystem, Diffie-Hellman key exchange and the Digital

Signature Algorithm. Current methods:

the Pohlig-Hellman algorithm if p-1 is a product of small primes, so this should be avoided in those applications


Methods

More sophisticated algorithms exist, usually inspired by similar algorithms for integer factorization. These algorithms run faster than the naive algorithm, but none of them runs in polynomial time. Baby-step giant-step (Also known as 'Little-Step Big-Step') Pollard's rho algorithm for logarithms Pollard's lambda algorithm (aka Pollard's kangaroo algorithm) Pohlig-Hellman algorithm Index calculus algorithm Number field sieve


Quadratic Residue

Quadratic Residue Integer b is a quadratic residue of modulo integer n if

and only if x2 b mod n has a solution for x Number x is called the square root of b Otherwise b is called quadratic nonresidue

Given odd prime p, b is quadratic residue, iff b(p-1)/2 1 mod p b is quadratic nonresidue, iff b(p-1)/2 -1 mod p These facts can be used to test primes with probability


Computing Square root mod p

Given number a, find number x, x2 =a mod p If p=3 mod 4, then x=a(p+1)/4 mod p is a solution. If p=5 mod 8, a(p-1)/4 =1 mod p then x= a(p+3)/8 mod p If p=5 mod 8, a(p-1)/4 =-1 mod p then x= 2a(4a)(p-5)/8 mod p If p=1 mod 8, x a N

hs k

1

2

p hk 1 2 Here h is an odd number


Compute square-root mod p

Find a solution to x2 =a mod p if exists Let r=0, s=p-1; while s even, {r=r+1; s=s/2;} Choose random n such that Let z=ns mod p; x=a(s+1)/2 mod p; b=as mod p; If b=1, return x as a solution Let m=1, y=b2 mod p; while y<>1 {y= y2 mod p; m=m+1;} If r=m then a is Quadratic non-residue; exit; Let x=xz2r-m-1 mod p and b=bz2r-m mod p and z=z2r-m mod p Go to step 4

The expected running time is O(log4 p)

n

p

1


Complexity Theory

The input length of a problem is the number n of symbols used to characterize it

Complexity of a method Function f(n) is order O(g(n)) if

f(n)<=c*|g(n)|, for all n>=N0, for some c Function f(n) is order (g(n)) if

f(n)>=c*|g(n)|, for all n>=N0, for some c Function f(n) is order (g(n)) if

c1*|g(n)|<=f(n)<=c2*|g(n)|, for all n>=N0, for some c1 and c2

Polynomial time algorithm (P) solves any instance of a particular problem with input length n in time

O(p(n)), where p is a polynomial


Cont.

Non-deterministic polynomial time algorithm (NP) is one for which any guess at the solution of an instance of

the problem may be checked for validity in polynomial time.

NP-complete problems are a subclass of NP problems for which it is known that if

any such problem has a polynomial time solution, then all NP problems have polynomial solutions.

Co-NP: the complements of NP problems.



Conventional Methods

Xiang-Yang Li


Roadmap of Cryptography

classical cryptography (--- 1920s) secret writing required only pen and paper Mostly: transposition, substitution ciphers Easily broken by statistics analysis (e.g., frequency)

mechanical devices invented for encryption Rotor machines (e.g. Enigma cipher) 1930s-1950s featured in films, such as in the James Bond adventure From

Russia with Love

specification of DES and the invention of RSA (1970s) --- modern ciphers Public key system, most notably

Quantum Cryptography (future?)


Quantum Cryptography Quantum cryptography currently has two aspects.

quantum key exchange (also known as quantum key distribution), a method for secure communications based on quantum mechanics

conjectured effect of quantum computing on cryptanalysis, although it is currently, like quantum computing itself, only a theoretical concept.

Basic idea of quantum key exchange is to use the "noisy" properties of light to render incoherent an image that acts to complement a secret key. This image can be represented in a number of ways, but the ability to decode

that image rests upon an understanding of how it was made. No way to intercept the transmission without changing it is possible, so key information can be exchanged with great confidence it has been transmitted secretly.

quantum computing will considerably extend the reach of cryptanalysis, making brute force key space searches much more effective -- if such computers ever become possible in actual practice


History

Ancient ciphers Have a history of at least 4000 years Ancient Egyptians enciphered some of their

hieroglyphic writing on monuments Ancient Hebrews enciphered certain words in the

scriptures 2000 years ago Julius Caesar used a simple substitution

cipher, now known as the Caesar cipher Roger bacon described several methods in 1200s


History

Ancient ciphers Geoffrey Chaucer included several ciphers in his works Leon Alberti devised a cipher wheel, and described the

principles of frequency analysis in the 1460s Blaise de Vigenère published a book on cryptology in

1585, & described the polyalphabetic substitution cipher

Increasing use, esp in diplomacy & war over centuries


Classical Cryptographic Techniques

Two basic components of classical ciphers: Substitution: letters are replaced by other letters Transposition: letters are arranged in a different order

These ciphers may be: Monoalphabetic: only one substitution/ transposition is

used, or Polyalphabetic:where several substitutions/ transpositions

are used

Product cipher: several ciphers concatenated together


Encryption and Decryption

Plaintextciphertext

Encipher C = E(K)(P) Decipher P = D(K)(C)

Key source


Key Management

Using secret channel Encrypt the key Third trusted party The sender and the receiver generate

key The key must be same We will talk more about how we can generate keys for

two parties who are “unknown” of each other before, and want secure communication


Attacks

Recover the message Recover the secret key

Thus also the message

Thus the number of keys possible must be large!


Possible Attacks

Ciphertext only Algorithm, ciphertext

Known plaintext Algorithm, ciphertext, plaintext-ciphertext pair

Chosen plaintext Algorithm, ciphertext, chosen plaintext and its ciphertext

Chosen ciphertext Algorithm, ciphertext, chosen ciphertext and its plaintext

Chosen text Algorithm, ciphertext, chosen plaintext and ciphertext


Steganography

Conceal the existence of message Character marking Invisible ink Pin punctures Typewriter correction ribbon

Cryptography renders message unintelligible!


Contemporary Equiv.

Least significant bits of picture frames 2048x3072 pixels with 24-bits RGB info Able to hide 2.3M message

Drawbacks Large overhead Virtually useless if system is known

Improvement Using some “random” sequence of the last bit for storing the data Challenge: produce such random sequence such that the attacker

cannot figure out the sequence!


Caesar Cipher

Replace each letter of message by a letter a fixed distance away Reputedly used by Julius Caesar

Example: L FDPH L VDZ L FRQTXHUHG

I CAME I SAW I CONGUERED

The mapping is ABCDEFGHIJKLMNOPQRSTUVWXYZ

DEFGHIJKLMNOPQRSTUVWXYZABC


Mathematical Model

Description Assume all letters are mapped to integers [0,25] A:-0, B-1, ….., Z25

Encryption E(k) : i i + k mod 26

Decryption D(k) : i i - k mod 26


Cryptanalysis: Caesar Cipher

Key space: 26Exhaustive key search

Example GDUCUGQFRMPCNJYACJCRRCPQ

HEVDVHRGSNQDOKZBDKDSSDQR Plaintext:

JGXFXJTIUPSFQMBDFMFUUFSTKHYGYKUJVGRNCEGNGVVGTU

Ciphertext: LIZHZLVKWRUHSODFHOHWWHUVMJAIAMWXSVITPEGIPIXXIVW


Character Frequencies

In most languages letters are not equally common in English e is by far the most common letter

Have tables of single, double & triple letter frequencies

Use these tables to compare with letter frequencies in ciphertext, a monoalphabetic substitution does not change relative

letter frequencies do need a moderate amount of ciphertext (100+ letters)


Letter Frequency Analysis

Single Letter A,B,C,D,E,…..

Double Letter TH,HE,IN,ER,RE,ON,AN,EN,….

Triple Letter THE,AND,TIO,ATI,FOR,THA,TER,RES,…


Letter Frequencies


Letter Frequencies


N-gram Frequencies

Digraph Frequency th he an in er on re ed nd ha at en es of nt ea ti to io

le is ou ar as de rt ve

Trigraph Frequency the and tha ent ion tio for nde has nce tis oft men

For more, see http://www.letterfrequency.org


Modular Arithmetic Cipher

Use a more complex equation to calculate the ciphertext letter for each plaintext letter

E(a,b) : i ai + b mod 26 Need gcd(a,26) = 1 Otherwise, not reversible So, a2, 13, 26 Caesar cipher: a=1, b=3


Cryptanalysis

Key space:12*26 Brute force search

Use letter frequency counts to guess a couple of possible letter mappings frequency pattern not produced just by a shift

But it is still a substitution, thus we can use frequency analysis

use these mappings to solve 2 simultaneous equations to derive above parameters


Playfair Cipher

The Playfair cipher or Playfair square is a manual symmetric encryption technique and was the first literal digraph substitution cipher. The scheme was invented in 1854 by Charles

Wheatstone, but bears the name of Lord Playfair who promoted the use of the cipher.


Playfair Cipher

s i/j m p l

e a b c d

f g h k n

o q r t u

v w x y z

Key: simple

Used in WWI and WWII


Playfair Cipher

Use filler letter to separate repeated letters

Encrypt two letters together Same row– followed letters

ac--bd Same column– letters under

qw--wi Otherwise—square’s corner at same row

ar--bq


Analysis

Size of diagrams: 25! But the actual different diagrams are not 25! Two diagrams are the same if they derive the same

encryption and decryption method Then what is the number of difference diagrams in

playfair cipher? 25!/25=24!

Difficult using frequency analysis But it still reveals the frequency information

Frequency of 2-gram (bi-gram, two-letters)


Playfair Cryptanalysis

Like most pre-modern era ciphers, the Playfair cipher can be easily cracked if there is enough text. Obtaining the key is relatively straightforward if both

plaintext and ciphertext are known. When only the ciphertext is known, brute force

cryptanalysis of the cipher involves searching through the key space for matches between the frequency of occurrence of digrams (pairs of letters) and the known frequency of occurrence of digrams in the assumed language of the original message.


Playfair, cont

A different approach to tackling a Playfair cipher is the shotgun hill climbing method. This starts with a random square of letters. Then minor changes

are introduced (i.e. switching letters, rows, or reflecting the entire square) to see if the candidate plaintext is more like standard plaintext than before the change (perhaps by comparing the trigrams to a known frequency chart).

If the new square is deemed to be an improvement, then it is adopted and then further mutated to find an even better candidate.

Eventually, the plaintext or something very close is found to achieve a maximal score by whatever grading method is chosen.

Computers can adopt this algorithm to crack Playfair ciphers with a relatively small amount of text.


Hill Cipher

Hill cipher is a polygraphic substitution cipher based on linear algebra. Invented by Lester S. Hill in 1929, it was the first polygraphic

cipher in which it was practical (though barely) to operate on more than three symbols at once.

Each letter is treated as a digit in base 26: A = 0, B =1, and so on. A block of n letters is then considered as a vector of n dimensions, and multiplied by a n × n matrix, modulo 26. The components of the matrix are the key, and should be random provided that the matrix is invertible in (to ensure decryption is possible).

The Hill cipher has achieved Shannon's diffusion, and an n-dimensional Hill cipher can diffuse fully across n symbols at once.

http://en.wikipedia.org/wiki/Confusion_and_diffusion


Hill Cipher Machine


Hill Cipher Machine

With fixed Key and patented Triple encryption was recommended for

security: a secret nonlinear step, followed by the wide diffusive

step from the machine, followed by a third secret nonlinear step.

Such a combination was actually very powerful for 1929, and indicates that Hill apparently understood the concepts of a meet-in-the-middle attack as well as confusion and diffusion.

Unfortunately, his machine did not sell.


Hill Cipher

Encryption Assign each letter an index C=KP mod 26 Matrix K is the key

Decryption P=K-1C mod 26 Thus, we can decrypt iff gcd(det(K), 26) =1.


How to Decrypt?

Compute K-1

Compute det(K) Check if gcd(det(K), 26) =1 If not, then K-1 do not exist Else K-1 is

1 1

1 1

1 1

1 1

1

1

1

1

2

1

K K

K K

K

n

n

n

n

n

n n

, ,

, ,

d et( )


cont

K

k k k k

k k k k

k k k k

k k k k

i j

j j n

i i j i j i n

i i i i

n n j n j n n

,

, , , ,

, , , ,

, , , ,

, , , ,

1 1 1 1 1 1 1

1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

1 1 1


Hill Cipher Cryptanalysis

Difficult to use frequency analysis But vulnerable to known-plaintext

attack Give simple method to attack hill cipher under the

known-plaintext assumption? How to attack under the chosen plaintext assumption?

The security could be greatly enhanced by combining with some non-linear step to defeat this attack.


Key Sizes

How may good keys? One might naïvely think that the key size, in bits, is n2log226 or

about 4.7n2. In fact, it is slightly less than this because not all

randomly selected matrices are usable. A slightly less naïve view might guess that 1/2 + 1/26 of candidate

keys would be unusable, reducing the keyspace by about 54%. In fact, determinants are not uniformly distributed, and

the key space reduction is closer to 70%. Additionally it seems to be prudent to avoid too many zeroes in

the key matrix, since they reduce diffusion. The net effect is that the effective keyspace of a basic

Hill cipher is about 4.64n2. For a 5 × 5 Hill cipher, that is about 114 bits. Of course,

key search is not the most efficient known attack

http://en.wikipedia.org/wiki/Bit


Polyalphabetic Substitution

Use more than one substitution alphabet

Makes cryptanalysis harder since have more alphabets to guess and flattens frequency distribution

same plaintext letter gets replaced by several ciphertext letter, depending on which alphabet is used


Vigenère Cipher

Basically multiple Caesar ciphers key is multiple letters long

K = k1 k2 ... kd

ith letter specifies ith alphabet to use use each alphabet in turn, repeating from start after d

letters in message

Plaintext THISPROCESSCANALSOBEEXPRESSED

Keyword CIPHERCIPHERCIPHERCIPHERCIPHE Ciphertext VPXZTIQKTZWTCVPSWFDMTETIGAHLH


Enigma Machine

Enigma was a portable cipher machine used to encrypt and decrypt secret messages. a family of related electro-mechanical rotor machines

German military

Japan commercial


Enigma Machine

Enigma encryption for two consecutive letters —

current is passed into set of rotors, around the reflector, and back out through the rotors again.

Letter A encrypts differently with consecutive key presses, first to G, and then to C. This is because the right hand rotor has stepped, sending the signal on a completely different route.


Enigma

the actual encipherment of a letter is performed electrically. When a key is pressed, the circuit is completed; current flows

through the various components and ultimately lights one of many lamps, indicating the output letter.

Current flows from a battery through the switch controlled by the depressed key into a fixed entry wheel. This leads into the rotor assembly (or scrambler), where the complex internal wiring of each rotor results in the current passing from one rotor to the next along a convoluted path. After passing through all the rotors, current enters the reflector, which relays the signal back out again through the rotors and the entry wheel — this time via a different path — and, finally, to one of the lamps (the earliest Enigma models do not have the reflector).


Rotors

performs a very simple type of encryption a simple substitution cipher


World War II Era Encryption Devices A few here

Sigaba (United States) Typex (Britain) Lorenz cipher (Germany) Geheimfernschreiber (Germany)

For more, see http://w1tp.com/enigma/


One-time Pad

theoretically unbreakable (Claude Shannon) the plaintext is combined with a random "pad" the same length as the

plaintext. Patent by

Gilbert Vernam (AT&T) and Joseph Mauborgne Encryption

C=PK Decryption

P=CK Claude Shannon's work can be interpreted as

that any information-theoretically secure cipher will be effectively equivalent to the one-time pad algorithm. Hence one-time pads offer the best possible mathematical security of any encryption scheme, anywhere and anytime.


One-time pad--cont Drawbacks

it requires secure exchange of the one-time pad material, which must be as long as the message

pad disposed of correctly and never reused In practice

Generate a large number of random bits, Exchange the key material securely between the users before sending

an one-time enciphered message, Keep both copies of the key material for each message securely until

they are used, and Securely dispose of the key material after use, thereby ensuring the

key material is never reused.

It requires a perfect random numbers as key We will learn how to generate pseudo-random numbers


Random numbers needed

If the key material is generated by a deterministic program then it is not actually random should never be used in an one-time pad cipher. If so used, the method becomes a stream cipher; these

usually employ a short key that is used to generate a long pseudorandom stream, which is then combined with the message using some such mechanism as those used in one-time pads. Stream ciphers can be secure in practice, but they cannot be absolutely secure in the same provable sense as the one-time pad


Stream ciphers

Stream ciphers The most famous: Vernam cipher Invented by Vernam, ( AT&T, in 1917) Process the message bit by bit (as a stream) different from the one-time pad– some call same Simply add bits of message to random key bits

Examples A well-known stream cipher is RC4; others include: A5/1, A5/2, Chameleon, FISH, Helix. ISAAC, Panama, Pike, SEAL, SOBER,

SOBER-128 and WAKE. Usage

Stream ciphers are used in applications where plaintext comes in quantities of unknowable length - for example, a secure wireless connection


Simplest Stream Cipher

Plaintext

Key

Ciphertext Ciphertext

Key

Plaintext


Pros and Cons

Drawbacks Need as many key bits as message, difficult in practice (ie distribute on a mag-tape or CDROM)

Strength Is unconditionally secure provided key is truly random


Key Generation

Why not to generate keystream from a smaller (base) key?Use some pseudo-random function to do this Although this looks very attractive, it proves to be very

very difficult in practice to find a good pseudo-random function that is cryptographically strong

This is still an area of much research


Transposition Methods

Permutation of plaintext Example

Write in a square in row, then read in column order specified by the key

Enhance: double or triple transposition Can reapply the encryption on ciphertext



Block CiphersXiang-Yang Li


Block Ciphers

The message is broken into blocks, Each of which is then encrypted (Like a substitution on very big characters - 64-bits or

more)


Substitution and Permutation

In his 1949 paper Shannon also introduced the idea of substitution-permutation (S-P) networks, which now form the basis of modern block ciphers An S-P network is the modern form of a substitution-

transposition product cipher S-P networks are based on the two primitive

cryptographic operations we have seen before


Substitution

A binary word is replaced by some other binary word

The whole substitution function forms the key

If use n bit words, The key space is 2n!

Can also think of this as a large lookup table, with n address lines (hence 2n addresses), each n bits wide being the output value

Will call them s-boxes


Cont.


Permutation

A binary word has its bits reordered (permuted)

The re-ordering forms the key If use n bit words,

The key space is n! (Less secure than substitution)

This is equivalent to a wire-crossing in practice (Though is much harder to do in software)

Will call these p-boxes


Cont.


Substitution-permutation Network Shannon combined these two primitives He called these mixing

transformations A special form of product ciphers where

S-boxes Provide confusion of input bits

P-boxes Provide diffusion across s-box inputs


Confusion and Diffusion

Confusion A technique that seeks to make the relationship

between the statistics of the ciphertext and the value of the encryption keys as complex as possible. Cipher uses key and plaintext.

Diffusion A technique that seeks to obscure the statistical

structure of the plaintext by spreading out the influence of each individual plaintext digit over many ciphertext digits.


Desired Effect

Avalanche effect A characteristic of an encryption algorithm in which a

small change in the plaintext gives rise to a large change in the ciphertext

Best: changing one input bit results in changes of approx half the output bits

Completeness effect

where each output bit is a complex function of all the input bits


Practical Substitution-permutation Networks In practice we need to be able to

decrypt messages, as well as to encrypt them, hence either: Have to define inverses for each of our S & P-boxes,

but this doubles the code/hardware needed, or Define a structure that is easy to reverse, so can use

basically the same code or hardware for both encryption and decryption


Feistel Cipher

Invented by Horst Feistel, working at IBM Thomas J Watson research labs in

early 70's,

The idea is to partition the input block into two halves, l(i-1) and r(i-1), use only r(i-1) in each round i (part) of the cipher

The function f incorporates one stage of the S-P network, controlled by part of the key k(i) known as the ith subkey


Cont.


Cont.

This can be described functionally as: L(i) = R(i-1) R(i) = L(i-1) f(k(i), R(i-1))

This can easily be reversed as seen in the above diagram, working backwards through the rounds

In practice link a number of these stages together (typically 16 rounds) to form the full cipher


Data Encryption Standard

Adopted in 1977 by the National Bureau of Standards, now the National Institute of Standards and Technology

Data are encrypted in 64-bit blocks using a 56-bit key

The same algorithm is used for decryption.

Subject to much controversy


History

IBM LUCIFER 60’s Uses 128 bits key

Proposal for NBS, 1973 Adopted by NBS, 1977

Uses only 56 bits key Possible brute force attack

Design of S-boxes was classified Hidden weak points in in S-Boxes?

Wiener (93) claim to be able to build a machine at $100,00 and break DES in 1.5 days


DES

DES encrypts 64-bit blocks of data, using a 56-bit key

the basic process consists of: an initial permutation (IP) 16 rounds of a complex key dependent calculation f a final permutation, being the inverse of IP Function f can be described as L(i) = R(i-1) R(i) = L(i-1) P(S( E(R(i-1)) K(i) ))


DES


Initial and Final Permutations

Inverse Permutations

40 8 48 16 56 24 64 32

39 7 47 15 55 23 63 31

38 6 46 14 54 22 62 30

37 5 45 13 53 21 61 29

36 4 44 12 52 20 60 28

35 3 43 11 51 19 59 27

34 2 42 10 50 18 58 26

33 1 41 9 49 17 57 25


Function f


Expansion Table

Expands the 32 bit data to 48 bits Result(i)=input( array(i))

32 1 2 3 4 5

4 5 6 7 8 9

8 9 10 11 12 13

12 13 14 15 16 17

16 17 18 19 20 21

20 21 22 23 24 25

24 25 26 27 28 29

28 29 30 31 32 1


S-Boxes

S-Box is a fixed 4 by 16 array Given 6-bits B=b1b2b3b4b5b6,

Row r=b1b6

Column c=b2b3b4b5

S(B)=S(r,c) written in binary of length 4


Example

S-Box S1

14

4 13

1 2 15

11

8 3 10

6 12

5 9 0 7

0 15

7 4 14

2 13

1 10

6 12

11

9 5 3 8

4 1 14

8 13

6 2 11

15

12

9 7 3 10

5 0

15

12

8 2 4 9 1 7 5 11

3 14

10

0 6 13


Permutation Table

The permutation after each round

16 7 20 21

29 12 28 17

1 15 23 26

5 18 31 10

2 8 24 14

32 27 3 9

19 13 30 6

22 11 4 25


Subkey Generation

Given a 64 bits key (with parity-check bit) Discard the parity-check bits Permute the remaining bits using fixed table P1 Let C0D0 be the result (total 56 bits)

Let Ci =Shifti(Ci-1); Di =Shifti(Di-1) and Ki

be another permutation P2 of CiDi (total 56 bits) Where cyclic shift one position left if i=1,2,9,16 Else cyclic shift two positions left


Permutation Tables

57

49

41

33

25

17

9

1 58

50

42

34

26

18

10

2 59

51

43

35

27

19

11

3 60

52

44

36

63

55

47

39

31

23

15

7 62

54

47

38

30

22

14

6 61

53

45

37

29

21

13

5 28

20

12

4

14

17

11

24

1 5

3 28

15

6 21

10

23

19

12

4 26

8

16

7 27

20

13

2

41

52

31

37

47

55

30

40

51

45

33

48

44

49

39

56

34

53

46

42

50

36

29

32

Permutation table P1 Permutation table P2


DES in Practice

DEC (Digital Equipment Corp. 1992) built a chip with 50k transistors Encrypt at the rate of 1G/second Clock rate 250 Mhz Cost about $300

Applications ATM transactions (encrypting PIN and so on)


Model

Mode of use The way we use a block cipher Four have been defined for the DES by ANSI in the

standard: ANSI X3.106-1983 modes of use)

Block modes Splits messages in blocks (ECB, CBC)

Stream modes On bit stream messages (CFB, OFB)


Block Modes

Electronic Codebook Book (ECB) where the message is broken into independent 64-bit

blocks which are encrypted Ci = DESK1 (Pi)

Cipher Block Chaining (CBC) again the message is broken into 64-bit blocks, but they

are linked together in the encryption operation with an IV

Ci = DESK1 (PiCi-1)

C-1=IV (initial value)


Stream Model

Cipher FeedBack (CFB) where the message is treated as a stream of bits, added

to the output of the DES, with the result being feed back for the next stage

Ci = Pi DESK1 (Ci-1)

C-1=IV (initial value)


Cont.

Output FeedBack (OFB) where the message is treated as a stream of bits, added

to the message, but with the feedback being independent of the message

Ci = Pi Oi

Oi = DESK1 (Oi-1)

O-1=IV (initial value)


DES Weak Keys

With many block ciphers there are some keys that should be avoided, because of reduced cipher complexity

These keys are such that the same sub-key is generated in more than one round, and they include:


Cont.

Weak keys The same sub-key is generated for every round DES has 4 weak keys

Semi-weak keys Only two sub-keys are generated on alternate rounds DES has 12 of these (in 6 pairs)

Demi-semi weak keys Have four sub-keys generated


Cont.

None of these causes a problem since they are a tiny fraction of all available keys

However they MUST be avoided by any key generation program


DES Attacks

1998:The EFF's US$250,000 DES cracking machine

contained 1,536 custom chips and could brute force a DES key in a

matter of days — the photo shows a DES Cracker

circuit board fitted with several Deep Crack chips.

http://en.wikipedia.org/wiki/Electronic_Frontier_Foundation

http://en.wikipedia.org/wiki/EFF_DES_cracker


DES Attacks:

The COPACOBANA machine, built for US$10,000 by the Universities of Bochum and Kiel, contains 120 low-cost FPGAs and can perform an exhaustive key search on DES in 9 days on average. The photo shows the backplane of the machine with the FPGAs


Attack Faster than Brute Force

Differential cryptanalysis was discovered in the late 1980s by Eli Biham and Adi Shamir,

although it was known earlier to both IBM and the NSA and kept secret. To break the full 16 rounds, differential cryptanalysis requires 247 chosen plaintexts. DES was designed to be resistant to DC.

Linear cryptanalysis was discovered by Mitsuru Matsui, and needs 243 known plaintexts

(Matsui, 1993); the method was implemented (Matsui, 1994), and was the first experimental cryptanalysis of DES to be reported. There is no evidence that DES was tailored to be resistant to this

type of attack.

http://en.wikipedia.org/wiki/Eli_Biham

http://en.wikipedia.org/wiki/Eli_Biham

http://en.wikipedia.org/wiki/Adi_Shamir



http://en.wikipedia.org/wiki/Chosen_plaintext

http://en.wikipedia.org/wiki/Mitsuru_Matsui

http://en.wikipedia.org/wiki/Known_plaintext


Possible Techniques for Improving DES Multiple enciphering with DES Extending DES to 128-bit data paths

and 112-bit keys Extending the key expansion calculation


Double DES?

Using two encryption stages and two keys C=Ek2(Ek1(P))

P=Dk1(Dk2(C))

It is proved that there is no key k3 such that C=Ek2(Ek1(P))=Ek3(P)

But Meet-in-the-middle attack


Meet-in-the-Middle Attack

Assume C=Ek2(Ek1(P)) Given the plaintext P and ciphertext C Encrypt P using all possible keys k1

Decrypt C using all possible keys k2

Check the result with the encrypted plaintext lists If found match, they test the found keys again for

another plaintext and ciphertext pair If it turns correct, then find the keys Otherwise keep decrypting C


Triple DES

DES variant Standardized in ANSI X9.17 & ISO 8732

and in PEM for key management Proposed for general EFT standard by

ANSI X9 Backwards compatible with many DES

schemes Uses 2 or 3 keys


Cont.

No known practical attacks Brute force search impossible (very

hard) Meet-in-the-middle attacks need 256

Plaintext-Ciphertext pairs per key Popular current alternative


IDEA:

Developed by James Massey & Xuejia Lai at ETH originally in Zurich in 1990, then called IPES: X Lai, J L Massey, "A Proposal for a New Block

Encryption Standard" in Advances in Cryptology - Eurocrypt '90, Lecture

Notes in Computer Science, vol 473, pp 389-404, X Lai, J L Massey, S Murphy, "Markov Ciphers and

Differential Cryptanalysis" in Advances in Cryptology - Eurocrypt '91, Lecture

Notes in Computer Science, vol 547, pp 17-38, name changed to IDEA in 1992


Basic Features

Encrypts 64-bit blocks using a 128-bit key Based on mixing operations from

different (incompatible) algebraic groups XOR, + mod 2^(16) , X mod 2^(16) +1) On 16-bit sub-blocks, with no permutations used

IDEA is patented in Europe & US, however non-commercial use is freely permitted used in the public domain PGP (with agreement) currently no attack against IDEA is known

Seem secure against differential cryptanalysis, brute force


Operations

Operations XOR, Addition mod 216, multiplication mod 216 +1

Why these special mod for addition, multiplication They do not satisfy the distributive law They do not satisfy the associative law


MA: multiplication/addition

Multiplication/addition Basic block to provide diffusion Input of MA

Two sub-blocks derived from 4 input sub-blocks, 4 sub-keys

Two other sub-keys Output

Two sub-blocks Needs four operations

Four operations are the minimum to provide full diffusion


Overview


Cont.

IDEA encryption works as follows: Use 8-rounds The 64-bit data is divided into: X1 , X2 , X3 , X4

Each round The sub-blocks are added (2,3), multiplied (1,4) with

sub-keys The results are XORed [1,3] and [2,4] to 2 sub-blocks The XOR results set as input of MA structure,

It outputs two subblocks Results are then XORed with 2,4 and 1,3 subblocks respectively

The second and third sub-blocks are swapped Finally new sub-keys are combined with the sub-blocks


Sub-Keys

Total need 52=6*8+4 sub-keys First are directly from key in order Left shift of 25 bits, and then next 8 sub-keys Each sub-key is a sub-block of the original key

Decryption Much more complicated It needs the inverse of the encryption key

For addition, multiplication


Decryption

The process of decryption is essentially the same as encryption But with different selection of sub-keys Basic Operations

K1.1^(-1 ) is the multiplicative inverse mod 2^(16) +1

-K1.2 is the additive inverse mod 2^(16) The original operations are:

(+) bit-by-bit XOR + additional mod 2^(16) of 16-bit integers * multiplication mod 2^(16) +1 (where 0 means 2^(16) )


Decryption Sub-Keys

Round Encryption Keys Decryption Keys 1 K1.1 K1.2 K1.3 K1.4 K1.5 K1.6 K9.1-1 -K9.2 -K9.3 K9.4-1

K8.5 K8.6 2 K2.1 K2.2 K2.3 K2.4 K2.5 K2.6 K8.1-1 -K8.3 -K8.2 K8.4-1

K7.5 K7.6 3 K3.1 K3.2 K3.3 K3.4 K3.5 K3.6 K7.1-1 -K7.3 -K7.2 K7.4-1

K6.5 K6.6 4 K4.1 K4.2 K4.3 K4.4 K4.5 K4.6 K6.1-1 -K6.3 -K6.2 K6.4-1

K5.5 K5.6 5 K5.1 K5.2 K5.3 K5.4 K5.5 K5.6 K5.1-1 -K5.3 -K5.2 K5.4-1

K4.5 K4.6 6 K6.1 K6.2 K6.3 K6.4 K6.5 K6.6 K4.1-1 -K4.3 -K4.2 K4.4-1

K3.5 K3.6 7 K7.1 K7.2 K7.3 K7.4 K7.5 K7.6 K3.1-1 -K3.3 -K3.2 K3.4-1

K2.5 K2.6 8 K8.1 K8.2 K8.3 K8.4 K8.5 K8.6 K2.1-1 -K2.3 -K2.2 K2.4-1

K1.5 K1.6 Output K9.1 K9.2 K9.3 K9.4 K1.1-1 -K1.2 -K1.3 K1.4-1


Important Feature

The size of the sub-block Need 216+1 be prime number

To compute the inverse for each possible subkey So sub-block size 8 is also possible

28+1=257 is prime number


CAST-128

By Carlisle Adams, Stafford Tavares Defined in RFC 2144 Use key size varying from 40 to 128 bits Structure of Feistel network 16 rounds on 64-bits data block Four primitive operations

Addition, substration (mod 232) Bitwise exclusive-OR Left-circular rotation


Skipjack and Clipper

Skipjack used in Clipper escrowed encryption scheme(US govt) Skipjack is a block cipher, 64-bit data hardware only implementation 80-bit key (escrowed in 2 halves) 32 round all design details and descriptions are classified has been very considerable debate over its use attack by Matt Blaze (ATT) on the LEAF component of

the Clipper protocol for secure phone communications


Blowfish Scheme

Developed by Bruce Schneier Fast, compact, simple and variably secure Two basic operations: addition, XOR Key ranges from 32 bits to 448 bits Similar to Feistel scheme The sub-key and s-boxes are complicated So not suitable when key changes often Function g is very simple, unlike DES


RC5

Developed by R. Rivest Suitable for hardware or software Fast, simple, low memory, data-dependent rotations Adaptable to processors of different word length

A family of algorithms determined by word length, number of rounds, size of secret key

Decryption and encryption are not the same With little variations

Primitive operations Addition, XOR, left circular rotation


Characteristics

Key features of advanced sym block cipher Variable key length Mixed operators Data dependent rotation Key dependent rotation Key dependent S-boxes Lengthy key schedule algorithm Variable function F Variable of number of rounds Operation on both halved data each round


AES

Advanced Encryption Standard (Rijndael) key size and the block size may be chosen to be any of 128, 192, or

256 bits (later only key, block fixed 128) Rijndael has a variable number of rounds. Not counting an extra round

performed at the end of encipherment with one step omitted, the number of rounds in Rijndael is:

9 if both the block and the key are 128 bits long. 11 if either the block or the key is 192 bits long, and

neither of them is longer than that. 13 if either the block or the key is 256 bits long.

Three big blocks first perform an Add Round Key step (XORing a subkey with

the block) by itself, then regular rounds noted above, the final round with the Mix Column step


Advanced Encryption Standard

a.k.a Lab #1

Not “American” Encryption Standard


How was AES created?

AES competition Started in January 1997 by NIST 4-year cooperation between

U.S. Government Private Industry Academia

Why? Replace 3DES Provide an unclassified, publicly disclosed encryption algorithm,

available royalty-free, worldwide


The Finalists

MARS IBM

RC6 RSA Laboratories

Rijndael Joan Daemen (Proton World International) and Vincent Rijmen (Katholieke Universiteit Leuven)

Serpent Ross Anderson (University of Cambridge), Eli Biham (Technion), and Lars Knudsen (University of California San Diego)

Twofish Bruce Schneier, John Kelsey, and Niels Ferguson (Counterpane, Inc.), Doug Whiting (Hi/fn, Inc.), David Wagner (University of California Berkeley), and Chris Hall (Princeton University)

Wrote the book on crypto


Evaluation Criteria (in order of importance)

Security Resistance to cryptanalysis, soundness of math,

randomness of output, etc.

Cost Computational efficiency (speed) Memory requirements

Algorithm / Implementation Characteristics Flexibility, hardware and software suitability,

algorithm simplicity


Results


Results


The winner: Rijndael

AES adopted a subset of Rijndael Rijndael supports more block and key sizes


Finite Fields

AES uses the finite field GF(28) b7x7 + b6x6 + b5x5 + b4x4 + b3x3 + b2x2 + b1x + b0

{b7, b6, b5, b4, b3, b2, b1, b0}

Byte notation for the element: x6 + x5 + x + 1 {01100011} – binary {63} – hex

Has its own arithmetic operations Addition Multiplication


Finite Field Arithmetic

Addition (XOR) (x6 + x4 + x2 + x + 1) + (x7 + x + 1) = x7 + x6 + x4 +

x2

{01010111} {10000011} = {11010100} {57} {83} = {d4}

Multiplication is tricky


Finite Field Multiplication ()

(x6 + x4 + x2 + x +1) (x7 + x +1) =

x13 + x11 + x9 + x8 + x7 + x7 + x5 + x3 + x2 + x + x6 + x4 + x2 + x +1

= x13 + x11 + x9 + x8 + x6 + x5 + x4 + x3 +1

and

x13 + x11 + x9 + x8 + x6 + x5 + x4 + x3 +1 modulo ( x8 + x4 + x3 + x +1) = x7 + x6 +1.

Irreducible Polynomial

These cancel


Efficient Finite field Multiply

There’s a better way xtime() – very efficiently multiplies its input by {02}

Multiplication by higher powers can be accomplished through repeat application of xtime()


Efficient Finite field Multiply

Example: {57} {13}{57} {02} = xtime({57}) = {ae}

{57} {04} = xtime({ae}) = {47}

{57} {08} = xtime({47}) = {8e}

{57} {10} = xtime({8e}) = {07}

{57} {13} = {57} ({01} {02} {10}) = ({57} {01}) ({57} {02}) ({57}

{10}) = {57} {ae} {07}

= {fe}


AES parameters

Nb – Number of columns in the State For AES, Nb = 4

Nk – Number of 32-bit words in the Key For AES, Nk = 4, 6, or 8

Nr – Number of rounds (function of Nb and Nk)

For AES, Nr = 10, 12, or 14


AES methods

Convert to state array Transformations (and their inverses)

AddRoundKey SubBytes ShiftRows MixColumns

Key Expansion


Convert to State Array

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Input block:

0 4 8 12

1 5 9 13

2 6 10 14

3 7 11 15

S0,0 S0,1 S0,2 S0,3

S1,0 S1,1 S1,2 S1,3

S2,0 S2,1 S2,2 S2,3

S3,0 S3,1 S3,2 S3,3

=


AddRoundKey

XOR each byte of the round key with its corresponding byte in the state array

S0,0 S0,1 S0,2 S0,3

S1,0 S1,1 S1,2 S1,3

S2,0 S2,1 S2,2 S2,3

S3,0 S3,1 S3,2 S3,3

S’0,0 S’0,1 S’0,2 S’0,3

S’1,0 S’1,1 S’1,2 S’1,3

S’2,0 S’2,1 S’2,2 S’2,3

S’3,0 S’3,1 S’3,2 S’3,3

S0,1

S1,1

S2,1

S3,1

S’0,1

S’1,1

S’2,1

S’3,1

R0,0 R0,1 R0,2 R0,3

R1,0 R1,1 R1,2 R1,3

R2,0 R2,1 R2,2 R2,3

R3,0 R3,1 R3,2 R3,3

R0,1

R1,1

R2,1

R3,1

XOR


SubBytes

Replace each byte in the state array with its corresponding value from the S-Box

00 44 88 CC

11 55 99 DD

22 66 AA EE

33 77 BB FF

55


ShiftRows

Last three rows are cyclically shifted

S0,0 S0,1 S0,2 S0,3

S1,0 S1,1 S1,2 S1,3

S2,0 S2,1 S2,2 S2,3

S3,0 S3,1 S3,2 S3,3

S1,0

S3,0 S3,1 S3,2

S2,0 S2,1


MixColumns

Apply MixColumn transformation to each column

S0,0 S0,1 S0,2 S0,3

S1,0 S1,1 S1,2 S1,3

S2,0 S2,1 S2,2 S2,3

S3,0 S3,1 S3,2 S3,3

S’0,0 S’0,1 S’0,2 S’0,3

S’1,0 S’1,1 S’1,2 S’1,3

S’2,0 S’2,1 S’2,2 S’2,3

S’3,0 S’3,1 S’3,2 S’3,3

S0,1

S1,1

S2,1

S3,1

S’0,1

S’1,1

S’2,1

S’3,1

MixColumns()S’0,c = ({02} S0,c) ({03} S1,c) S2,c S3,c

S’1,c = S0,c ({02} S1,c) ({03} S2,c) S3,c

S’2,c = S0,c S1,c ({02} S2,c ) ({03} S3,c)

S’3,c = ({03} S0,c) S1,c S2,c ({02} S3,c


Key Expansion

Expands the key material so that each round uses a unique round key Generates Nb(Nr+1) words

Filled with just the key

Filled with a combination of the previous work and

the one Nk positions earlier


Encryption

byte state[4,Nb]

state = in

AddRoundKey(state, keySchedule[0, Nb-1])

for round = 1 step 1 to Nr–1 {SubBytes(state)ShiftRows(state) MixColumns(state)AddRoundKey(state, keySchedule[round*Nb, (round+1)*Nb-1])

}

SubBytes(state)ShiftRows(state)AddRoundKey(state, keySchedule[Nr*Nb, (Nr+1)*Nb-1])

out = state

First and last operations involve the key

Prevents an attacker from even beginning to encrypt or

decrypt without the key


Decryption

byte state[4,Nb]

state = in

AddRoundKey(state, keySchedule[Nr*Nb, (Nr+1)*Nb-1])

for round = Nr-1 step -1 downto 1 {InvShiftRows(state) InvSubBytes(state)AddRoundKey(state, keySchedule[round*Nb, (round+1)*Nb-1])InvMixColumns(state)

}

InvShiftRows(state)InvSubBytes(state)AddRoundKey(state, keySchedule[0, Nb-1])

out = state


Encrypt and Decrypt

Encryption

AddRoundKey

SubBytesShiftRowsMixColumnsAddRoundKey

SubBytesShiftRowsAddRoundKey

Decryption

AddRoundKey

InvShiftRowsInvSubBytesAddRoundKeyInvMixColumns

InvShiftRowsInvSubBytesAddRoundKey



Public key systemXiang-Yang Li


Public Key Encryption

Two difficult problems Key distribution under conventional encryption Digital signature

Diffie and Hellman, 1976 Astonishing breakthrough One key for encryption and the other related key for

decryption It is computationally infeasible to determine the

decryption key using only the encryption key and the algorithm


Public Key Cryptosystem

Essential steps of public key cryptosystem Each end generates a pair of keys

One for encryption and one for decryption Each system publishes one key, called public key, and

the companion key is kept secret It A wants to send message to B

Encrypt it using B’s public key When B receives the encrypted message

It decrypt it using its own private key


Applications of PKC

Encryption/Decryption The sender encrypts the message using the receiver’s

public key Q: Why not use the sender’s secret key?

Digital signature The sender signs a message by encrypt the message or

a transformation of the message using its own private key

Key exchange Two sides cooperate to exchange a session key,

typically for conventional encryption


Conditions of PKC

Computationally easy To generate public and private key pair To encrypt the message using encryption key To decrypt the message using decryption key

Computational infeasible To compute the private key using public key To recover the plaintext using ciphertext and public key

The encryption and decryption can be applied in either order


One Way Function

PKC boils down to one way function Maps a domain into a range with unique inverse The calculation of the function is easy The calculation of the inverse is infeasible

Easy The problem can be solved in polynomial time

Infeasible The effort to solve it grows faster than polynomial time For example: 2n

It requires infeasible for all inputs, not just worst case


Trapdoor One-way Function

Trapdoor one way function Maps a domain into a range with unique inverse

Y=fk(X)

The calculation of the function is easy The calculation of the inverse is infeasible if the key is

not known The calculation of the inverse is easy if the key is

known


Possible Attacks

Brute force Use large keys

Trade-off: speed (not linearly depend on key size) Confined to small data encryption: signature, key

management

Compute the private key from public key Not proven that is not feasible for most protocols!

Probable message attack Encrypt all possible messages using encryption key Compare with the ciphertext to find the matched one! If data is small, feasible, regardless of key size of PKC


History

http://www.research.att.com/~smb/nsam-160/

British National Security Action

Memorandum 160 Kennedy Nuclear Weapon http://www.research.att.com/~smb/nsam-160/pg1.html


RSA Algorithm

R. Rivest, A. Shamir, L. Adleman (1977) James Ellis came up with the idea in 1970, and proved that it was

theoretically possible. In 1973, Clifford Cocks a British mathematician invented a variant on RSA; a few months later, Malcom Williamson invented a Diffie-Hellman analog

Only revealed till 1997 Patent expired on September 20, 2000. Block cipher using integers 0~n-1

Thus block size k is less than log2n Algorithm:

Encryption: C=Me mod n Decryption: M=Cd mod n

Both sender and the receiver know n


RSA (public key encryption)

Alice wants Bob to send her a message. She: selects two (large) primes p, q, TOP SECRET, computes n = pq and (n) = (p-1)(q-1),

(n) also TOP SECRET, selects an integer e, 1 < e < (n), such that

gcd(e, (n)) = 1, computes d, such that de 1 (mod (n)),

d also TOP SECRET, gives public key (e, n), keeps private key (d,

n).


Requirements

Possible to find e and d such that M=Mde mod n for all message M

Easy to conduct encryption and decryption

Infeasible to compute d Given n and e


RSA Example

1. Select primes: p=17 & q=112. Compute n = pq =17×11=1873. Compute ø(n)=(p–1)(q-1)=16×10=1604. Select e : gcd(e,160)=1; choose e=75. Determine d: de=1 mod 160 and d < 160

Value is d=23 since 23×7=161= 10×160+16. Publish public key KU={7,187}7. Keep secret private key KR={23,17,11}


RSA Example cont

sample RSA encryption/decryption is: given message M = 88 (nb. 88<187) encryption:

C = 887 mod 187 = 11

decryption:M = 1123 mod 187 = 88


Key Generation

Recall Euler Theorem a(n)+1 =a mod n for all 0<a<n and gcd(a,n)=1 Then ed=1 mod (n) is sufficient to make algorithm

correct (need more proofs)

RSA chooses the following Integer n=pq for two primes p and q Select e, such that gcd(e, (n))=1 Compute the inverse of e mod (n)

The result is set as d


Key Generation

The prime numbers p and q must be sufficiently large They are chosen by applying primality testing of

randomly chosen large numbers About n/ln n prime numbers less than n

Implies needs to check about 2ln n random numbers to find 2 primes numbers around n

Compute n=pq, keep p and q secret!

Select random number e Test gcd(e, (n))=1, and get d if equation holds


Exponentiation

can use the Square and Multiply Algorithm a fast, efficient algorithm for exponentiation concept is based on repeatedly squaring base and multiplying in the ones that are needed to

compute the result look at binary representation of exponent only takes O(log2 n) multiples for number n

eg. 75 = 74.71 = 3.7 = 10 mod 11 eg. 3129 = 3128.31 = 5.3 = 4 mod 11


Exponentiation


More on Exponention (PGP)

To compute Cd mod n, we compute Cd mod p and Cd mod q

Remember that the receiver could keep p,q

Then Chinese Remainder Theorem to find Cd mod n


Security of RSA

Brute force: try all possible private keys Factoring integer n, then know (n)

Not proven to be NPC

Determine (n) directly without factoring Equivalent to factoring! (1996)

Determine d directly without knowing (n) Currently appears as hard as factoring

But not proven, so it may be easier!


Practical Considerations Testing p, q using probability first, then deterministic methods A good random number generator is needed for p,q

'random' and 'unpredictable' Primes p and q should be in similar scale Both p-1 and q-1 should have large prime factor The gcd(p-1,q-1) should be small The encryption key e = 2 should not be used The decryption key d should larger then n1/4

RSA is much slower than symmetric cryptosystems. In practice, typically encrypts a secret message with a symmetric algorithm, encrypts the (comparatively short)

symmetric key with RSA, and transmits both the RSA-encrypted symmetric key and the symmetrically-encrypted message to Alice.


Fixed point of RSA

How many m such that me=m mod n assume that gcd(m, n)=1 It is same as me-1=1 mod n Thus, me-1=1 mod p and me-1=1 mod q Solutions gcd(e-1,p-1)*gcd(e-1,q-1)

Need more proofs.


Cyclic Attack

Compute me mod n, me2 mod n, me3 mod n…till it reaches some message readable.

Need period large Let r be the largest prime of p-1, L be the

largest prime of r-1 Then period is at least L with high probability

Implies that we often need find a large prime x Based on this, find a large prime of y=kx+1 format (by trying

k=2,3,…) Based on y, then find a large prime p=t y+1 format

Try difference values for t=2,3,4…


How to deal with p, q

Delete them securely Or used for speed-up calculation from

CRT Compute Me mod p and Me mod q Then find using Me mod n based on CRT


Timing Attacks

Keep track of how long a computer takes to decrypt a message! Paul Kocher, 1995, Dec-7 Stunning attack strategy and cipher only attack! Guessing the key bit by bit

Countermeasures (Rivest 11 Dec 1995) Constant exponentiation time Random delay Blinding (add a random number for encryption and

decryption)


Chosen Ciphertext Attack

Collect ciphertext c (send to Alice), want to find m=cd mod n

Attacker chooses random r Compute x= re mod n; y=xc mod n; and t= r-1

mod n Attacker gets Alice to sign y with private key

using RSA: yd mod n That is why not use the same key for encryption and digital signature

Alice sends u= yd mod n to Attacker Attacker then computes tu mod nm


Other attacks on RSA

Comprised decryption key If the private key d (for decryption of received

ciphertext) of a user is comprised, then the user has to reselect n and e and d

It cannot use the old number n to produce the key-pairs! Otherwise attacker already can factor n almost surely!

The number n can only be used by one person If two user uses the same n, even they do not know the

factoring of n, they still could figure out the factoring of n with probability almost one.

Similar as above


Bit security of RSA

Given ciphertext C, We may want to find the last bit of M, denoted by

parity(C) We may want to find if M>n/2, denoted by half(C) We may want to find all bits of M

The above three attacks are the same! If we can solve one, we can solve the other two!


Other Public Key Systems

Rabin Cryptosystem Decryption is not unique

Elgamal Cryptosystem Expansion of the plaintext (double)

Knapsack System Already broken

Elliptic Curve System If directly implement Elgamal on elliptic curve

Expansion of plaintext by 4; Restricted plaintext Menezes-Vanston system is more efficient


Rabin Cryptosystem

Procedure Let n=pq and p=3 mod 4, q=3 mod 4 Publish n, and a number b<n For message m

C=m(m+b) mod n The receiver decrypts ciphertext C

(b2/4+C)1/2-b/2


Analysis

For receiver, need solve equation x2+xb=C mod n Let x1=x+b/2, c=b2/4+C, then need

Solve x12 =c mod n

Chinese Remainder Theorem implies that x1

2 =c mod p

x12 =c mod q

When p=3 and q=3 mod 4 Solution x1=c(p+1)/4 mod p and x1=c(q+1)/4 mod q

Then Chinese Remainder Theorem again to combine solution


Security

Breaking it < factoring n Secure against

Chosen plaintext attack Not secure against

Chosen ciphertext attack

Decoding produces three false results in addition to the correct one, so that the correct result must be guessed. This is the major disadvantage of the Rabin cryptosystem and one of the factors which have prevented it from finding widespread practical use.

It has been proven that decoding the Rabin cryptosystem is equivalent to the integer factorization problem, which is rather different than for RSA.


Dealing with 4 solutions

By adding redundancies, for example, the repetition of the last 64 bits, the system can be made to produce a single root.

If this technique is applied, the proof of the equivalence with the factorization problem fails.


ElGamal Cryptosystem

Based on Discrete Logarithm Find unique integer a such that gx=y mod p

Here x is a primitive element in Zp, p is prime

Procedure Make p, g, y public, keep x secret Encryption:

Ek(m)=(gk mod p, m y k mod p)

Decryption Dk(y1,y2)=y2(y1

x)-1 mod p


Security of ElGamal

ElGamal is a simple example of a semantically secure asymmetric key encryption algorithm (under reasonable assumptions).

ElGamal's security rests, in part, on the difficulty of solving the discrete logarithm problem in G. Specifically, if the discrete logarithm problem could be solved

efficiently, then ElGamal would be broken. However, the security of ElGamal actually relies on the so-called Decisional Diffie-Hellman (DDH) assumption. This assumption is often stronger than the discrete log assumption, but is still believed to be true for many classes of groups.


Semantic Security

Semantic security is a widely-used definition for security in an PKS. For a cryptosystem to be semantically secure, it must be infeasible

for a computationally-bounded adversary to derive significant information about a message (plaintext) when given only its ciphertext and the corresponding public encryption key.

Semantic security considers only the case of a "passive" attacker, i.e., one who observes ciphertexts and generates chosen ciphertexts using the public key

Indistinguishability definition is used more commonly than the original definition of semantic security.


Indistinguishability: semantic security. Indistinguishability under Chosen Plaintext Attack (IND-CPA) is

commonly defined by the following game: A probabilistic polynomial time-bounded adversary is given a public key, which it

may use to generate any number of ciphertexts (within polynomial bounds). The adversary generates two equal-length messages m0 and m1, and transmits them

to a challenge oracle along with the public key. The challenge oracle selects one of the messages by flipping a uniformly-weighted

coin, encrypts the message under the public key, and returns the resulting ciphertext c to the adversary.

The underlying cryptosystem is IND-CPA (and thus semantically secure under chosen plaintext attack) if

the adversary cannot determine which of the two messages was chosen by the oracle, with probability significantly greater than 1 / 2 (the success rate of random guessing).

a semantically secure encryption scheme must by definition be probabilistic, possessing a component of randomness; if this were not the case, the adversary could simply compute the deterministic encryption of m0 and m1 and compare these encryptions with the returned ciphertext c to successfully guess the oracle's choice.


Deal with deterministic PKS

RSA, can be made semantically secure (under stronger assumptions) through the use of random encryption padding schemes such as Optimal Asymmetric Encryption Padding (OAEP).

ElGamal scheme is semantically secure


Bit security of Discrete Log

Given gx=y mod p We may want to find the value of x Find some bits of x

Assume that p-1 = 2st We can find the last s bits of x for sure But to find the other bits of x is same as to find all bits

of x!

Example, the last bit of x is 0 y is QR iff y(p-1)/2=1 mod p 1 y is NQR iff y(p-1)/2=-1 mod p


DH Assumption

Consider a cyclic group G of order q. The DDH assumption states that, given (g,ga,gb) for a randomly-chosen generator g and random ,

the value gab "looks like" a perfectly random element of G. This intuitive notion is formally stated by

saying that the following two ensembles are computationally indistinguishable: (g,ga,gb,gab), where g,a,b are chosen at random as

described above (this input is called a "DDH tuple"); (g,ga,gb,gc), where g,a,b are chosen at random and c

is chosen at random. Diffie-Hellman problem

computing gab from (g,ga,gb)


Knapsack Cryptosystem

Based on subset sum problem Given a set, find a subset with half summation value It is NPC problem generally

Superincreasing set if si>j<isj

The subset problem over superincreasing set can be solved in polynomial time!

Been broken by Shamir, 1984 Using integer programming tech by Lenstra


Solve Subset Problem

Let T be the half summation, t=T; For i=n downto 1 do

If tsi then t=t-si

Set xi=1

Else xi=0

If xisi=T then (x1, x2,… xn) is the solution Else, there is no solution


Knapsack System

Procedure Select a superincreasing set s Let p be prime larger than set summation of s, Select integer a, keep s, a, p secret Make t=(as1, as2,…asn) mod p public

Encryption Ciphertext C = E(x1,x2,…xn)=xiti mod p

Decryption Solve the subset summation problem (s, a-1C mod

p)


Elliptic Curve Cryptography

majority of public-key crypto (RSA, D-H) use either integer or polynomial arithmetic with very large numbers/polynomials

imposes a significant load in storing and processing keys and messages

an alternative is to use elliptic curves offers same security with smaller bit

sizes


Real Elliptic Curves

an elliptic curve is defined by an equation in two variables x & y, with coefficients

consider a cubic elliptic curve of form y2 = x3 + ax + b where x,y,a,b are all real numbers also define zero point O

have addition operation for elliptic curve geometrically sum of Q+R is reflection of intersection R


Real Elliptic Curve Example


Finite Elliptic Curves

Elliptic curve cryptography uses curves whose variables & coefficients are finite

have two families commonly used: prime curves Ep(a,b) defined over Zp

use integers modulo a prime p best in software

binary curves E2m(a,b) defined over GF(2n) use polynomials with binary coefficients best in hardware


Elliptic Curve Cryptography

ECC addition is analog of modulo multiply ECC repeated addition is analog of modulo

exponentiation need “hard” problem equiv to discrete log

Q=kP, where Q,P belong to a prime curve is “easy” to compute Q given k,P but “hard” to find k given Q,P known as the elliptic curve logarithm problem

Certicom example: E23(9,17)


ECC Diffie-Hellman

can do key exchange analogous to D-H users select a suitable curve Ep(a,b)

select base point G=(x1,y1) with large order n s.t. n*G=O

A & B select private keys nA<n, nB<n

compute public keys: PA=nA×G, PB=nB×G

compute shared key: K=nA×PB, K=nB×PA

same since K=nA×nB×G


ECC Encryption/Decryption

several alternatives, will consider simplest must first encode any message M as a

point on the elliptic curve Pm

select suitable curve & point G as in D-H each user chooses private key nA<n and computes public key PA=nA×G to encrypt Pm : Cm={kG, Pm+k PA}, k

random decrypt Cm compute:

Pm+kPA–nA(kG) = Pm+k(nAG)–nA(kG) = Pm


ECC Security

relies on elliptic curve logarithm problem

fastest method is “Pollard rho method” compared to factoring, can use much

smaller key sizes than with RSA etc for equivalent key lengths computations

are roughly equivalent hence for similar security ECC offers

significant computational advantages


Cryptography and Network

Key Management and generation

Xiang-Yang Li


Key Exchange

Public key systems are much slower than private key system Public key system is then often for short data

Signature, key distribution

Key distribution One party chooses the key and transmits it to other user

Key agreement Protocol such two parties jointly establish secret key

over public communication channel Key is the function of inputs of two users


Distribution of Public Keys

can be considered as using one of: Public announcement Publicly available directory Public-key authority Public-key certificates


Public Key Management

Simple one: publish the public key Such as newsgroups, yellow-book, etc. But it is not secure, although it is convenient

Anyone can forge such a announcement Ex: user B pretends to be A, and publish a key for

A Then all messages sent to A, readable by B!

Let trusted authority maintain the keys Need to verify the identity, when register keys User can replace old keys, or void old keys


Possible Attacks

Observe all messages over the channel So assume that all plaintext messages are available to

all

Save messages for reuse later So have to avoid replay attack

Masquerade various users in the network So have to be able to verify the source of the message


Public Announcement

users distribute public keys to recipients or broadcast to community at large eg. append PGP keys to email messages or post to news

groups or email list

major weakness is forgery anyone can create a key claiming to be someone else

and broadcast it until forgery is discovered can masquerade as claimed

user


Publicly Available Directory

can obtain greater security by registering keys with a public directory

directory must be trusted with properties: contains {name,public-key} entries participants register securely with directory participants can replace key at any time directory is periodically published directory can be accessed electronically

still vulnerable to tampering or forgery


Public-Key Authority

improve security by tightening control over distribution of keys from directory

has properties of directory and requires users to know public key

for the directory then users interact with directory to

obtain any desired public key securely does require real-time access to directory when keys

are needed


Public-Key Authority


Cont.

More advanced distribution A sends request-for-key(B) to authority with time-

stamp, that is, Ida|Idb|Time Authority replies with key(B) (encrypted by its private

key), that is EKTta(KUb| Ida|Idb|Time)

A initiates a message to B, including a random number Na, its IDA

B then ask authority to get key(A) B sends A (encrypted by A’s public key) Na and Nb

A then replies B Nb encrypted by B’s public key


Cont.

In above scheme, the authority is bottleneck

New approach: certificate Any user can read certificate, determine name and

public key of the certificate’s owner Any user can verify the authority of certificate Only the authority can create and update certificate Any user can verify the time-stamp of certificate

The certificate is CA=EKRauth[T,IDA, KUA] Time-stamp is to avoid reuse of voided key


Public-Key Certificates

certificates allow key exchange without real-time access to public-key authority

a certificate binds identity to public key usually with other info such as period of validity, rights of use etc

with all contents signed by a trusted Public-Key or Certificate Authority (CA)

can be verified by anyone who knows the public-key authorities public-key

To validate the certificate, we need another certificate, one that matches the Issuer (of CA) in the first certificate. Then we take the RSA public key from the second (CA) certificate, use it to decode the signature on the first certificate to obtain an MD5 hash, which must match an actual MD5 hash computed over the rest of the certificate.


X.509

The structure of a X.509 v3 digital certificate is as follows: Certificate

Version Serial Number Algorithm ID Issuer Validity

Not Before Not After

Subject Subject Public Key Info

Public Key Algorithm Subject Public Key

Issuer Unique Identifier (Optional) Subject Unique Identifier (Optional) Extensions (Optional)

... Certificate Signature Algorithm Certificate Signature

http://en.wikipedia.org/wiki/Digital_certificate


Sample Certificate Certificate: Data: Version: 1 (0x0) Serial Number: 7829 (0x1e95) Signature Algorithm: md5WithRSAEncryption Issuer: C=ZA, ST=Western Cape, L=Cape Town, O=Thawte Consulting cc, OU=Certification

Services Division, CN=Thawte Server CA/[email protected] Validity

Not Before: Jul 9 16:04:02 1998 GMT Not After : Jul 9 16:04:02 1999 GMT

Subject: C=US, ST=Maryland, L=Pasadena, O=Brent Baccala, OU=FreeSoft, CN=www.freesoft.org/[email protected]

Subject Public Key Info: Public Key Algorithm: rsaEncryption RSA Public Key: (1024 bit) Modulus (1024 bit): 00:b4:31:98:0a:c4:bc:62:c1:88:aa:dc:b0:c8:bb:

33:35:19:d5:0c:64:b9:3d:41:b2:96:fc:f3:31:e1: 66:36:d0:8e:56:12:44:ba:75:eb:e8:1c:9c:5b:66: 70:33:52:14:c9:ec:4f:91:51:70:39:de:53:85:17: 16:94:6e:ee:f4:d5:6f:d5:ca:b3:47:5e:1b:0c:7b: c5:cc:2b:6b:c1:90:c3:16:31:0d:bf:7a:c7:47:77: 8f:a0:21:c7:4c:d0:16:65:00:c1:0f:d7:b8:80:e3: d2:75:6b:c1:ea:9e:5c:5c:ea:7d:c1:a1:10:bc:b8: e8:35:1c:9e:27:52:7e:41:8f

Exponent: 65537 (0x10001) Signature Algorithm: md5WithRSAEncryption

93:5f:8f:5f:c5:af:bf:0a:ab:a5:6d:fb:24:5f:b6:59:5d:9d: 92:2e:4a:1b:8b:ac:7d:99:17:5d:cd:19:f6:ad:ef:63:2f:92: ab:2f:4b:cf:0a:13:90:ee:2c:0e:43:03:be:f6:ea:8e:9c:67: d0:a2:40:03:f7:ef:6a:15:09:79:a9:46:ed:b7:16:1b:41:72: 0d:19:aa:ad:dd:9a:df:ab:97:50:65:f5:5e:85:a6:ef:19:d1: 5a:de:9d:ea:63:cd:cb:cc:6d:5d:01:85:b5:6d:c8:f3:d9:f7: 8f:0e:fc:ba:1f:34:e9:96:6e:6c:cf:f2:ef:9b:bf:de:b5:22: 68:9f


Security

In 2005, Arjen Lenstra and Benne de Weger demonstrated "how to use hash collisions to construct two X.509 certificates that contain identical signatures and that differ only in the public keys," achieved using a collision attack on the MD5 hash function

See http://www.win.tue.nl/~bdeweger/

CollidingCertificates/ddl-full.pdf

http://en.wikipedia.org/wiki/Arjen_Lenstra



http://en.wikipedia.org/w/index.php?title=Benne_de_Weger&action=edit

http://en.wikipedia.org/w/index.php?title=Benne_de_Weger&action=edit

http://en.wikipedia.org/wiki/MD5


Public-Key Certificates


Public-Key Distribution of Secret Keys

use previous methods to obtain public-key

can use for secrecy or authentication but public-key algorithms are slow so usually want to use private-key

encryption to protect message contents hence need a session key have several alternatives for

negotiating a suitable session


Simple Secret Key Distribution

proposed by Merkle in 1979 A generates a new temporary public key pair A sends B the public key and their identity B generates a session key K sends it to A encrypted

using the supplied public key A decrypts the session key and both use

problem is that an opponent can intercept and impersonate both halves of protocol


Secret key Distribution

Simple secret key distribution A generates KUA and KRA, sends KUA to B

B generates a secret key ks

B sends ks to A using A’s public key KUA

A decrypts the message to get the secret key ks

To get more security, the public/private keys can be regenerated when needed

But vulnerable to the active attack! Attacker E can compromise the communication

between A and B as follows


Cont.

Attacking A generates KUA and KRA, sends IDA, KUA to B

E intercepts the message, transmits IDA, KUE to B

B generates a secret key ks

B sends ks to A using A’s “public key” KUE

E intercepts the message, decrypt it and get ks

E sends A the message Ks, encrypted by KUA

A decrypts the message to get the secret key ks

Now E knows Ks, but A, B are unaware of it


Secret Key Distribution

So need confidentiality and authentication A and B need to use a secure method to exchange their

public keys

Schemes A initiates a message to B, EKUB(Na,IDa)

B replies it with EKUA(Na,Nb)

A then replies it with EKUB(Nb)

A sends B the message EKUB (EKRA(Ks))

Security The first 3 steps are used to assure that A is A, B is B


Public-Key Distribution of Secret Keys

if have securely exchanged public-keys:


Key Predistribution

Trusted Authority (TA) generates keys for all pair of users and transmits to them Large overhead (for TA and user)

Blom Scheme Keys are chosen from a finite field Zp

P is public prime number TA transmits k+1 elements of Zp to each user over

secure channel Secure condition: any set of at most k users (not U,V)

can not determine any information about Ku,v


Blom Scheme

Scheme (when k=1) Each user u has distinct element ru from Zp

TA choose a,b,c and defines f(x,y)=a+b(x+y)+cxy mod p

For each u, TA computes gu(x)=f(x, ru) mod p

TA transmits gu(x) to user u

Two users u and v compute the common key f(ru, rv)= a+b(ru + rv)+c ru rv mod p

Here f(ru, rv)= gv(ru)= gu(rv)


Security of Blom Scheme

Less than k users can not determine keys

However, more than k users can compute any keys Solving equations to get a,b,c for k=1

Generally Function f(x,y)=Sum ai,jxiyj mod p

Here ai,j=aj,i


Diffie-Hellman Key Predist.

Computationally secure if discrete logarithm is intractable

Scheme Assume prime number p public and an integer c public Each user u has secret component au

User u computes bu=c au mod p

TA certifies it by computing (ID(u), bu, sigTA(ID(u), bu))

The common key of two users u and v is K=c

au av mod p


Diffie Hellman

Around September 1974, Diffie (Graduate student) had been traveling USA with his wife, Mary, discussing cryptography with anyone who was available. At the time, there was very little published

material about modern methods and much was classified. Very few people were interested in the topic and Marty Hellman even says that many of his colleagues felt that it was "born classified," like secrets about the atomic bomb, because it was so important to national security.

John Gill gave the idea of exponential


Diffie-Hellman Problem

Diffie-Hellman problem definition Given bu=gau mod p, bv=gav mod p, how to compute

gavau mod p? Here g is a primitive element of mod p The problem is not harder than the discrete log-

arithmetic problem, because the later one can always be used to solve it

It can be proved that it has the same difficulty as the ElGamal encryption system


Diffie-Hellman Key Exchange

Computationally secure if discrete logarithm is intractable

Scheme Assume prime number p public and an integer c public Each user u chooses a secret component au (new!)

User u computes bu=c au mod p

User v computes bv=c av mod p


au av mod p


Middle Attack

Intruder w intercept the communications Intruder w communications with u Intruder w communications with v The key computed by u is

K=c au av’ mod p

u w vc

au c au’

c av’ c

av


Authenticated Key Agreement

Introducing the identification scheme before key exchange does not help The attacker remains inactive until identification done

Simplified station to station protocol Key agreement protocol itself authenticates the user’s

identity at the same time the key being defined


Station-to-station Protocol

Scheme Each user has a certificate

C(v)=(Idv,verv,sigTA(Idv,verv)) User u selects au and computes bu=c

au mod p User v selects av and computes

Value bv=c av mod p

Key K=c au av mod p

Signature yv=sigv(bu,bv) User v sends (C(V), bv, yv) to U User u computes K=c

au av mod p, verifies yv, and C(V) User u computes yu=sigu(bu,bv), sends (C(u),yu) to V User v verifies yu, and C(u)


MTI Agreement Protocol

Scheme Assume prime number p public and an integer c public Each user has certificate c(u)=(Idu,bu, sigTA(Idu,bu))

Here bu= c au mod p

Each user u chooses a secret component ru (new!)

User u computes su=c ru mod p, sends (c(u),su)

User v computes sv=c rv mod p, sends (c(v),sv)


rvau+ ru av mod p= sv aubv ru mod p= su

avbu rv mod p



Authentication

Xiang-Yang Li


Message Authentication Digital Signature Authentication

Authentication requirements Authentication functions

Mechanisms MAC: message authentication code Hash functions, security in hash functions Hash and MAC algorithms

MD5, SHA, RIPEMD-160, HMAC

Digital signatures


Message Attacks

Possible attacks Disclosure Traffic analysis Masquerade Content modification Sequence modification Time modification Repudiation

Denial of the receipt of message by the destination or

Denial of the transmitting by the source


Authentication

Enables receiver to verify message authenticity Using some lower level functions as primitive

Three types of functions Message encryption Message authentication code (MAC) Hash function


Authentication

Goal: Bob wants Alice to “prove” her identity to him

Protocol ap1.0: Alice says “I am Alice”

Failure scenario??“ I am Alice”


Authentication

Goal: Bob wants Alice to “prove” her identity to him

Protocol ap1.0: Alice says “I am Alice”

in a network,Bob can not “see”

Alice, so Trudy simply declares

herself to be Alice“ I am Alice”


Authentication: another try

Protocol ap2.0: Alice says “I am Alice” in an IP packetcontaining her source IP address

Failure scenario??

“ I am Alice”Alice’s

IP address



Protocol ap2.0: Alice says “I am Alice” in an IP packetcontaining her source IP address

Trudy can createa packet

“spoofing”Alice’s address“ I am Alice”

Alice’s IP address



Protocol ap3.0: Alice says “I am Alice” and sends her secret password to “prove” it.

Failure scenario??

“ I’m Alice”Alice’s IP addr

Alice’s password

OKAlice’s IP addr



Protocol ap3.0: Alice says “I am Alice” and sends her secret password to “prove” it.

playback attack: Trudy records Alice’s

packetand later

plays it back to Bob


Alice’s password

OKAlice’s IP addr


Alice’s password


Authentication: yet another try

Protocol ap3.1: Alice says “I am Alice” and sends her encrypted secret password to “prove” it.

Failure scenario??


encrypted password

OKAlice’s IP addr



Protocol ap3.1: Alice says “I am Alice” and sends her encrypted secret password to “prove” it.

recordand

playbackstill works!


encryptedpassword

OKAlice’s IP addr


encryptedpassword


Authentication: yet another try

Goal: avoid playback attack

drawbacks?

Nonce: number (R) used only once –in-a-lifetime

ap4.0: to prove Alice “live”, Bob sends Alice nonce, R. Alice

must return R, encrypted with shared secret key“ I am Alice”

R

K (R)A-B

Alice is live, and only Alice knows key to encrypt

nonce, so it must be Alice!


Authentication: ap5.0

ap4.0 requires shared symmetric key can we authenticate using public key techniques?ap5.0: use nonce, public key cryptography

“ I am Alice”

RBob computes

K (R)A-

“ send me your public key”

K A+

(K (R)) = RA

-K A

+

and knows only Alice could have the

private key, that encrypted R such that

(K (R)) = RA-

K A+


ap5.0: security holeMan (woman) in the middle attack: Trudy poses

as Alice (to Bob) and as Bob (to Alice)

I am Alice I am Alice

R

TK (R)

-

Send me your public key

TK

+A

K (R)-

Send me your public key

AK

+

TK (m)+

Tm = K (K (m))+

T-

Trudy gets

sends m to Alice encrypted

with Alice’s public key

AK (m)+

Am = K (K (m))+

A-

R


ap5.0: security holeMan (woman) in the middle attack: Trudy poses

as Alice (to Bob) and as Bob (to Alice)

Difficult to detect: Bob receives everything that Alice sends, and vice versa. (e.g., so Bob, Alice can meet one week later and recall conversation) problem is that Trudy receives all messages as well!


Message Encryption

Conventional Encryption Authentication provided due to the secret key But the message need to be meaningful

What happened it message is not readable? How to determine intelligible automatically?

Approach Checksum or frame check sequence(FCS) to message Encrypt the message and the appending FCS Receiver decrypt the ciphertext Computes FCS of message, compare with received one


Public Key Encryption

Direct encryption by receiver’s public key Only confidentiality, no authentication

For authentication Encrypt using sender’s private key Assume the message is intelligible No confidentiality: everyone can decrypt

Confidentiality and authentication Encrypt by sender’s, then receiver’s public key But too time-consuming: 4 rounds RSA on large data


Message Authentication Code

Assume both uses share secret key k Procedure

Sender computes MAC=Ck(M) for M

Sent M and MAC of it to receiver Receiver computes the MAC on received M Compare it with received MAC If match, then accepts the message

MAC is similar to encryption, but not need be reversible!


MAC with Confidentiality

Two options Using another key to encrypt M and MAC Using another key to encrypt M only

Requirements of MAC Size of MAC: n Size of key: k Need 2n computations of MAC and n/k pairs of Mi and

MACi


Why not Conventional Encrypt

Possible situations Broadcast a message (one destination can verify) Authentication is done selectively Authentication of computer program Authentication may be important than secrecy Architecture flexibility Authentication lasts longer than secret protection


MAC Requirements

Computationally infeasible to construct M’ such that Ck(M’)=Ck(M)

Ck(M) uniformly distributed


Data Authentication Algorithm

ANSI standard X9.17 Based on DES Using Cipher Block Chaining mode

Data is grouped into 64 bits blocks Padding 0’s if necessary

Outputi=Ek(DiOutputi-1) 0<i, and Output0=0’s

The data authentication code DAC consists of the leftmost m bits of the last output, m16


Authentication Protocols

Central issues Confidentiality: prevent masqueraded and

compromised Timeliness: prevent replay attacks

Simple replay, repetition within timestamp, replay arrives but not the true messages,backward replay attack to the sender

Mutual authentication One-way authentication


Coping with Replay

Time stamps Party A accepts a message only if has valid timestamp

within a valid time Need synchronized clock How to set the synchronized clock?

Network delay consideration?

Challenge/response Party A, (receiver), sends B a nonce (challenge) and

requires the subsequent message contains it


Challenge-Response

To ensure a password is never sent in the clear. Given a client and a server share a key server sends a random challenge vector client encrypts it with private key and returns this server verifies response with copy of private key can repeat protocol in other direction to authenticate

server to client (2-way authentication) Secret key management

physically distributed before secure communications keys are stored in a central trusted key server


Conventional Encryption App.

Each user shares a secret master key with KDC (Key Distribution Center) Kerberos is an example Needham-Schroeder protocol Party A KDC Ida|Idb|Na KDCA Eka(Ks|Idb|Na|Ekb(Ks|Ida))

AB Ekb(Ks|Ida)

BA Eks(Nb)

AB Eks(f(Nb))


Analysis

Step 4 and 5 prevent the replay of step 3 Assume that Ks is not compromised

If Ks is compromised Vulnerable to replay attack Attacker can replay step 3 Unless B remembers all previous session keys with A,

it can not tell that it is a replay!


Denning Protocol

Denning Protocol Party A KDC Ida|Idb KDCA Eka(Ks|Idb|T|Ekb(Ks|Ida|T))

AB Ekb(Ks|Ida|T)

BA Eks(Nb)

AB Eks(f(Nb))

Here T is timestamp assures the freshness of the key Ks Rely on synchronized clock


Public-key Encryption App.

The simple one proposed by Denning AS: authentication server AAS Ida|Idb

ASA Ekras(KUa|Ida|T)|Ekras

(Kub|Idb|T)

AB Ekras(KUa|Ida|T)|Ekras

(Kub|Idb|T)|

Ekub(Ekra(Ks|T))

It needs clock synchronization


Cont.

Protocol by Woo and Lam, using nonce AKDC Ida|Idb KDCA EKRau(Idb|KUb)

AB EKUb(Na|Ida)

BKDC Idb|Ida|EKUau(Na)

KDCB EKRau(Ida|KUa)|EKUb(EkRau(Na|Ks|Ida|Idb))

BA EKUa(EkRau(Na|Ks|Ida|Idb) | Nb)

AB Eks(Nb)


One-way Authentication

Using Public Key approach If confidentiality is main concern

AB: EKUb(Ks) | Eks(M)

If authentication is main concern AB: M|EKRa(H(M))

This can not avoid the interception and replay attack

Sign the message then EKUb(M|EKRa(H(M)) )

Or EKUb(Ks) | Eks(M|EKRa(H(M)) )

Also A can sends the digital certificate EKRau(T|Ida|KUa)


Authentication Applications

will consider authentication functions developed to support application-level

authentication & digital signatures will consider Kerberos – a private-key

authentication service then X.509 directory authentication

service


Kerberos

Trusted key server system developed by MIT Provides centralized third-party authentication in a

distributed network access control may be provided for

each computing resource in either a local or remote network (realm)

A Key Distribution Centre (KDC), containing database: principles (customers and services) encryption keys

KDC provides non-corruptible authentication credentials (tickets or tokens)


Kerberos

Two Kerberos versions 4 : restricted to a single realm 5 : allows inter-realm authentication, in beta test Kerberos v5 is an Internet standard specified in RFC1510

To use Kerberos need to have a KDC on your network need to have Kerberised applications running on all participating

systems

US export restrictions Cannot be directly distributed outside US in source format

Crypto libraries must be re-implemented locally


Kerberos Requirements

first published report identified its requirements as: security reliability transparency scalability

implemented using an authentication protocol based on Needham-Schroeder


Kerberos 4 Overview

a basic third-party authentication scheme

have an Authentication Server (AS) users initially negotiate with AS to identify self AS provides a non-corruptible authentication credential

(ticket granting ticket TGT)

have a Ticket Granting server (TGS) users subsequently request access to other services

from TGS on basis of users TGT


Kerberos 4 Overview


Kerberos Realms

a Kerberos environment consists of: a Kerberos server a number of clients, all registered with server application servers, sharing keys with server

this is termed a realm typically a single administrative domain

if have multiple realms, their Kerberos servers must share keys and trust


Kerberos Version 5

developed in mid 1990’s provides improvements over v4

addresses environmental shortcomings encryption alg, network protocol, byte order,

ticket lifetime, authentication forwarding, interrealm auth

and technical deficiencies double encryption, non-std mode of use, session

keys, password attacks

specified as Internet standard RFC 1510


Authentication Protocols

used to convince parties of each others identity and to exchange session keys

may be one-way or mutual key issues are

confidentiality – to protect session keys timeliness – to prevent replay attacks


Replay Attacks

where a valid signed message is copied and later resent simple replay repetition that can be logged repetition that cannot be detected backward replay without modification

countermeasures include use of sequence numbers (generally impractical) timestamps (needs synchronized clocks) challenge/response (using unique nonce)


Using Symmetric Encryption

as discussed previously can use a two-level hierarchy of keys

usually with a trusted Key Distribution Center (KDC) each party shares own master key with KDC KDC generates session keys used for connections

between parties master keys used to distribute these to them


Needham-Schroeder Protocol

original third-party key distribution protocol

for session between A B mediated by KDC

protocol overview is:1. A→KDC: IDA || IDB || N1

2. KDC→A: EKa[Ks || IDB || N1 || EKb[Ks||IDA] ]

3. A→B: EKb[Ks||IDA]

4. B→A: EKs[N2]

5. A→B: EKs[f(N2)]


Needham-Schroeder Protocol

used to securely distribute a new session key for communications between A & B

but is vulnerable to a replay attack if an old session key has been compromised then message 3 can be resent convincing B that is

communicating with A

modifications to address this require: timestamps (Denning 81) using an extra nonce (Neuman 93)


Using Public-Key Encryption

have a range of approaches based on the use of public-key encryption

need to ensure have correct public keys for other parties

using a central Authentication Server (AS)

various protocols exist using timestamps or nonces


Denning AS Protocol

Denning 81 presented the following:1. A→AS: IDA || IDB

2. AS→A: EKRas[IDA||KUa||T] || EKRas[IDB||KUb||T]

3. A→B: EKRas[IDA||KUa||T] || EKRas[IDB||KUb||T] || EKUb[EKRas[Ks||T]]

note session key is chosen by A, hence AS need not be trusted to protect it

timestamps prevent replay but require synchronized clocks


One-Way Authentication

required when sender & receiver are not in communications at same time (eg. email)

have header in clear so can be delivered by email system

may want contents of body protected & sender authenticated


Using Symmetric Encryption

can refine use of KDC but can’t have final exchange of nonces, vis:1. A→KDC: IDA || IDB || N1

2. KDC→A: EKa[Ks || IDB || N1 || EKb[Ks||IDA] ]

3. A→B: EKb[Ks||IDA] || EKs[M]

does not protect against replays could rely on timestamp in message, though email

delays make this problematic


Public-Key Approaches

have seen some public-key approaches if confidentiality is major concern, can use:

A→B: EKUb[Ks] || EKs[M]

has encrypted session key, encrypted message

if authentication needed use a digital signature with a digital certificate:A→B: M || EKRa[H(M)] || EKRas[T||IDA||KUa]

with message, signature, certificate


Differences between Authentication and Digital Signature Two authentications:

Data authentication is comparable to stamping a document in a way disallowing all future modifications to it. Data authentication is usually accompanied with

data origin authentication that bounds a concrete person to this document Digital signature is a cryptographic technique that enables to

protect digital information (represented as a bit-stream) from undesirable modification. Since signature cannot just be appended to a digital bitstream, more sophisticated methods (also known as signatures schemes) for signing have been elaborated.

Signature scheme is a function Sig of a key pair (SA,VA) and a bitstring M, such that

for anyone who knows the secret key SA, it is easy to compute for any plaintext M the signature C=Sig(PA,M).

for anyone who knows VA (the public key), C and M, it is easy to verify if C=Sig(SA,M).

for a randomly chosen C, it is intractable for anyone who does not know SA to find a value M for which C=Sig(SA,M).



Hash Algorithms

Xiang-Yang Li


Hash Function

Map a message to a smaller value Requirements

Be applied to a block of data of any size Produced a fixed length output H(x) is easy to compute (by hardware, software) One-way: given code h, it is computationally infeasible

to find x: H(x)=h Weak collision resistance: given x, computationally

infeasible to find y so H(x)=H(y) Strong collision resistance: Computationally

infeasible to find x, y so H(x)=H(y)


Hash Algorithms

see similarities in the evolution of hash functions & block ciphers increasing power of brute-force attacks leading to evolution in algorithms from DES to AES in block ciphers from MD4 & MD5 to SHA-1 & RIPEMD-160 in hash

algorithms

likewise tend to use common iterative structure as do block ciphers


Basic Uses of Hash Function

Six basics usages Ek(M||H(M))

Confidentiality and authentication M|| Ek(H(M))

Authentication M|| EKRa(H(M))

Authentication and digital signature Ek(M|| EKRa(H(M)))

Authentication, digital signature and confidentiality M||H(M||S)

Authentication (S shared by both sides) Ek(M||H(M||S))

Confidentiality and authentication


Birthday Attacks

If 64-bits hash code is used On average, how many messages need to try to find one

match the intercepted hash code?

Birthday paradox A will sign a message appended with m-bits hash code Attacker generates some variations of fraud message,

also variations of good message Find pair of message each from the two sets messages

Such that they have the same hash code Give good message to A to get signature Replace good message with fraud message


Analysis

Using birthday attack, given 64-bits hash code How many message variations needed so the success

probability is large, say 90%?


Examples

Simple hash functions XOR of the input message

H(M)=X1 X2 … Xm-1 Xm

But not secure Ym=H(M) Y1 Y2 … Ym-1 has same hash value

as (X1X2 … Xm-1 Xm), where Yi is any value


Cont.

Based on DES, block chaining technique Rabin, 1978 Divide message M into fix-sized blocks Mi

Assume total n data blocks

H0=initial value

Hi=Emi[Hi-1]

Hn is the hash value

Birthday attack still applies If still 64-bits code used


More Attacks

Birthday attack applied if chosen plaintext

Meet in the middle attack if known plaintext Known signed hash code G Construct n-2 desired message block Qi

Compute Hi=EQi[Hi-1]

Generate 2m/2 random blocks X For each X, Compute Hn-1=EX[Hn-2]

Generate 2m/2 random blocks Y For each Y, Compute H’n-1=DY[G]

Find X, Y such that Hn-1= H’n-1

Then Q1, Q2,…Qn-2, X,Y is a fraud message


Security

The size of hash code determines security 128bits is not secure Currently, most use 160 bits hash code

Now recommend 256 bits

Attack MAC Objective is to find valid (x, Ck(x)) pair

Attack the key space: roughly 2k, k =key size Attack the MAC value


More Hash Algorithms

Algorithms Message Digest:MD5 (was mostly widely used) Secure Hash Algorithm: SHA-1 (from MD4) RIPEMD-160 HMAC


MD5

designed by Ronald Rivest (the R in RSA)

latest in a series of MD2, MD4 produces a 128-bit hash value until recently was the most widely used

hash algorithm in recent times have both brute-force & cryptanalytic

concerns

specified as Internet standard RFC1321


MD5 Overview

1. pad message so its length is 448 mod 512 2. append a 64-bit length value to message 3. initialise 4-word (128-bit) MD buffer

(A,B,C,D) 4. process message in 16-word (512-bit)

blocks: using 4 rounds of 16 bit operations on message block &

buffer add output to buffer input to form new buffer value

5. output hash value is the final buffer value


MD5 Overview


MD5 Compression Function

each round has 16 steps of the form: a = b+((a+g(b,c,d)+X[k]+T[i])<<<s)

a,b,c,d refer to the 4 words of the buffer, but used in varying permutations note this updates 1 word only of the buffer after 16 steps each word is updated 4 times

where g(b,c,d) is a different nonlinear function in each round (F,G,H,I)

T[i] is a constant value derived from sin


MD5 Compression Function


MD4

precursor to MD5 also produces a 128-bit hash of

message has 3 rounds of 16 steps vs 4 in MD5 design goals:

collision resistant (hard to find collisions) direct security (no dependence on "hard" problems) fast, simple, compact favours little-endian systems (eg PCs)


Strength of MD5

MD5 hash is dependent on all message bits

Rivest claims security is good as can be known attacks are:

Berson 92 attacked any 1 round using differential cryptanalysis (but can’t extend)

Boer & Bosselaers 93 found a pseudo collision (again unable to extend)

Dobbertin 96 created collisions on MD compression function (but initial constants prevent exploit)

conclusion is that MD5 looks vulnerable soon


Bad news

Chinese authors (Wang, Feng, Lai, and Yu) reported a family of collisions in MD5 (fixing the previous bug in their analysis), and also

reported that their method can efficiently (2^40 hash steps) find a collision in SHA-0.

August Crypto 2004,

MD5 is fatally wounded; its use will be phased out. SHA-1 is still alive but the vultures are circling. A gradual transition away from SHA-1 will now start. The first stage will be a debate about alternatives, leading to a consensus among practicing cryptographers about what the substitute will be.


Why collisions are bad

An example of what you might do with this. You could request an SSL certificate (for your real identity)

from a certificate authority. After the response comes back, you can then use that response (which is based on the MD5 of your identity+key) to "authenticate" a carefully chosen different certificate, one which claims that you are LargeBankOrSoftwareCorp., but which has the same MD5 as your real identity. You can then present this to other people in order to convince them that you are someone whom you are not.

Another example, core internet routers use md5 to exchange passwords. I

simply sniff the md5sum, and if I can find a string that generates the same sum, easily, I can send my own routing update that takes down the internet. More examples, since a LOT of applications use md5, but you get the idea.


Further detail

Obviously the above attack isn't quite so simple, but this research makes it *possible*. Before, it was believed to be sufficiently difficult to find a collision, that nobody worried about it. Now they are saying its feasible to do it in hours.

The question hanging around right now is that these researchers managed to find collisions easily, but not for an artbitrary string. The questions is how long before someone modifies this method to find any colllision. That is how much time the world has to move away.

More at http://www.freedom-to-tinker.com/archives/000664.html


What to do next

The U.S. National Institute of Standards and Technology is having a competition for a new cryptographic hash function.

The phrase "one-way hash function" might sound arcane and geeky, but hash functions are the workhorses of modern cryptography.

Submissions will be due in fall 2008, and a single standard is scheduled to be chosen by the end of 2011.

we have an interim solution in SHA-256.

http://csrc.nist.gov/pki/HashWorkshop/timeline.html


Secure Hash Algorithm (SHA-1)

SHA was designed by NIST & NSA in 1993, revised 1995 as SHA-1

US standard for use with DSA signature scheme standard is FIPS 180-1 1995, also Internet RFC3174 nb. the algorithm is SHA, the standard is SHS

produces 160-bit hash values now the generally preferred hash

algorithm based on design of MD4 with key

differences


SHA Overview1. pad message so its length is 448 mod

512 2. append a 64-bit length value to message3. initialise 5-word (160-bit) buffer

(A,B,C,D,E) to (67452301,efcdab89,98badcfe,10325476,c3d2e1f0)

4. process message in 16-word (512-bit) chunks: expand 16 words into 80 words by mixing & shifting use 4 rounds of 20 bit operations on message block &

buffer add output to input to form new buffer value



SHA-1 Compression Function

each round has 20 steps which replaces the 5 buffer words thus:(A,B,C,D,E) <-(E+f(t,B,C,D)+(A<<5)+Wt+Kt),A,

(B<<30),C,D)

a,b,c,d refer to the 4 words of the buffer t is the step number f(t,B,C,D) is nonlinear function for

round Wt is derived from the message block Kt is a constant value derived from sin


SHA-1 Compression Function


SHA-1 verses MD5

brute force attack is harder (160 vs 128 bits for MD5)

not vulnerable to any known attacks (compared to MD4/5)

a little slower than MD5 (80 vs 64 steps) both designed as simple and compact optimised for big endian CPU's (vs MD5

which is optimised for little endian CPU’s)


Revised Secure Hash Standard

NIST have issued a revision FIPS 180-2 adds 3 additional hash algorithms SHA-256, SHA-384, SHA-512 designed for compatibility with

increased security provided by the AES cipher

structure & detail is similar to SHA-1 hence analysis should be similar


RIPEMD-160

RIPEMD-160 was developed in Europe as part of RIPE project in 96

by researchers involved in attacks on MD4/5 initial proposal strengthen following analysis

to become RIPEMD-160 somewhat similar to MD5/SHA uses 2 parallel lines of 5 rounds of 16 steps creates a 160-bit hash value slower, but probably more secure, than SHA


RIPEMD-160 Overview

1. pad message so its length is 448 mod 512 2. append a 64-bit length value to message3. initialise 5-word (160-bit) buffer (A,B,C,D,E)

to (67452301,efcdab89,98badcfe,10325476,c3d2e1f0)

4. process message in 16-word (512-bit) chunks:

use 10 rounds of 16 bit operations on message block & buffer – in 2 parallel lines of 5

add output to input to form new buffer value



RIPEMD-160 Round


RIPEMD-160 Compression Function


RIPEMD-160 Design Criteria

use 2 parallel lines of 5 rounds for increased complexity

for simplicity the 2 lines are very similar step operation very close to MD5 permutation varies parts of message

used circular shifts designed for best results


RIPEMD-160 verses MD5 & SHA-1

brute force attack harder (160 like SHA-1 vs 128 bits for MD5)

not vulnerable to known attacks, like SHA-1 though stronger (compared to MD4/5)

slower than MD5 (more steps) all designed as simple and compact SHA-1 optimised for big endian CPU's vs

RIPEMD-160 & MD5 optimised for little endian CPU’s


Keyed Hash Functions as MACs

have desire to create a MAC using a hash function rather than a block cipher because hash functions are generally faster not limited by export controls unlike block ciphers

hash includes a key along with the message

original proposal:KeyedHash = Hash(Key|Message) some weaknesses were found with this

eventually led to development of HMAC


HMAC

specified as Internet standard RFC2104 uses hash function on the message:

HMACK = Hash[(K+ XOR opad) ||

Hash[(K+ XOR ipad)||M)]]

where K+ is the key padded out to size and opad, ipad are specified padding

constants overhead is just 3 more hash calculations than

the message needs alone any of MD5, SHA-1, RIPEMD-160 can be used


HMAC Overview


HMAC Security

know that the security of HMAC relates to that of the underlying hash algorithm

attacking HMAC requires either: brute force attack on key used birthday attack (but since keyed would need to observe

a very large number of messages)

choose hash function used based on speed verses security constraints


Summary

have considered: some current hash algorithms: MD5, SHA-1, RIPEMD-

160 HMAC authentication using hash function



Digital Signature

Xiang-Yang Li


Digital Signature


Digital Signatures

have looked at message authentication but does not address issues of lack of trust

digital signatures provide the ability to: verify author, date & time of signature authenticate message contents be verified by third parties to resolve disputes

hence include authentication function with additional capabilities


Digital Signature Properties

must depend on the message signed must use information unique to sender

to prevent both forgery and denial

must be relatively easy to produce must be relatively easy to recognize & verify be computationally infeasible to forge

with new message for existing digital signature with fraudulent digital signature for given message

be practical save digital signature in storage


Securities

A total break results in the recovery of the signing key.

A universal forgery attack results in the ability to forge signatures for any message.

A selective forgery attack results in a signature on a message of the adversary's choice.

An existential forgery merely results in some valid message/signature pair not already known to the adversary.


Classification of Digital Signature Undeniable Fail-Stop Blind One-time Multi-party (group signature) (n,k)-multi-party Oblivious Multi-undeniable


Algorithm and legal concerns several prior requirements

quality algorithms. Some public key algorithms are known to be insecure, practicable attacks against them having been identified.

quality implementations. An implementation of a good algorithm with mistake(s) will not work. (about 1 defect per 1,000 lines).

the private key must remain actually secret; if it becomes known to some other party, that party can produce perfect digital signatures of anything whatsoever.

distribution of public keys must be done in such a way that the public key claimed to belong to Bob actually belongs to Bob, and vice versa. This is commonly done using a public key infrastructure and the public key user association is attested by the operator of the PKI (called a certificate authority). For 'open' PKIs in which anyone can request such an attestation, the possibility of mistake is non trivial.

users (and their software) must carry out the signature protocol properly. Legal concerns


Direct Digital Signatures

involve only sender & receiver assumed receiver has sender’s public-key digital signature made by sender signing

entire message or hash with private-key can encrypt using receivers public-key important that sign first then encrypt

message & signature security depends on sender’s private-key


Arbitrated Digital Signatures

involves use of arbiter A validates any signed message then dated and sent to recipient

requires suitable level of trust in arbiter can be implemented with either private

or public-key algorithms arbiter may or may not see message


RSA signature

N=p q,where p and q are large primes Alice’s private key (e,n), Alice’s public key (d,n)

Signature of message m by Alice S=H(m)e mod n

Verification of signature by Bob Check if h(m) = Sd mod n


From wikipedia


Cont.

Typically d is chosen small (3 or 216+1) Problem:

Easy to create the signature of h(m1)h(m2)

RSA-PSS Use some more randomization to enhance security It was added in version 2.1 of PKCS #1 (see RFC 3447

).

http://en.wikipedia.org/wiki/PKCS

http://tools.ietf.org/html/rfc3447


ElGamal Signature

Global public components Prime number p with 512-1024 bits Primitive element g in Zp

Users private key Random integer x less than p

Users public key Integer y=gx mod p


Elgamal

Signature For each message M, generates random k Computes r=gk mod p Computes s=k-1(H(M)-xr) mod (p-1) Signature is (r,s)

Verifying Computes v1=gH(M) mod p

Computes v2=yrrs mod p

Test if v1= v2


Proof of Correctness

Computes v2=yrrs mod q So v2=yrrs mod q =gxr gks mod p

= gxr+k k-1(H(M)-xr) mod (p-1) mod p =gH(M) mod p=v1

Notice that here it uses Fermat theorem to show That g (H(M)-xr) mod (p-1) mod p = g (H(M)-xr) mod p


Cont.

The main disadvantage of ElGamal is the need for randomness (sometimes it is good), and its slower speed (especially for signing). Another potential disadvantage of the ElGamal system

is that message expansion by a factor of two takes place during encryption. However, such message expansion is negligible if the cryptosystem is used only for exchange of secret keys.


Digital Signature Standard

FIPS PUB 186 by NIST, 1991 Final announcement 1994 It uses

Secure Hashing Algorithm (SHA) for hashing Digital Signature Algorithm (DSA) for signature The hash code is set as input of DSA The signature consists of two numbers

DSA Based on the difficulty of discrete logarithm Based on Elgamal and Schnorr system


DSA

Global public components Prime number p with 512-1024 bits Prime divisor q of (p-1) with 160 bits Integer g=h(p-1)/q mod p

Users private key Random integer x less than q

Users public key Integer y=gx mod p


DSA

Signature For each message M, generates random k Computes r=(gk mod p) mod q Computes s=k-1(H(M)+xr) mod q Signature is (r,s)

Verifying Computes w=s-1 mod q, u1=H(M)w mod q

Computes u2=rw mod q,v=(gu1yu2 mod p) mod q

Test if v=r


Proof of Correctness

Notice that v=(gu1yu2 mod p) mod q =(gH(M)w mod q yrw mod q mod p) mod q =(gH(M)w mod q gxrw mod q mod p) mod q =(gH(M)w +xrw mod q mod p) mod q =(g(H(M)+xr)w mod q mod p) mod q =(g(H(M)+xr)k(H(M)+xr)-1 mod q mod p) mod q =(gk mod p) mod q =r


In practice (Sun Java Library) g = F7E1A085D69B3DDE CBBCAB5C36B857B9 7994AFBBFA3AEA82 F9574C0B3D078267 5159578EBAD4594F E67107108180B449 167123E84C281613 B7CF09328CC8A6E1 3C167A8B547C8D28 E0A3AE1E2BB3A675 916EA37F0BFA2135 62F1FB627A01243B CCA4F1BEA8519089 A883DFE15AE59F06 928B665E807B5525 64014C3BFECF492A

p = FD7F53811D751229 52DF4A9C2EECE4E7 F611B7523CEF4400 C31E3F80B6512669 455D402251FB593D 8D58FABFC5F5BA30 F6CB9B556CD7813B 801D346FF26660B7 6B9950A5A49F9FE8 047B1022C24FBBA9 D7FEB7C61BF83B57 E7C6A8A6150F04FB 83F6D3C51EC30235 54135A169132F675 F3AE2B61D72AEFF2 2203199DD14801C7

q = 9760508F15230BCC B292B982A2EB840B F0581CF5 Here g and p have 1024 bits, while q has 160 bits. They fulfill the requirement

that gq = 1 mod p,


Note

Can we use the random number k twice? What will happen if k used twice? We have r=(gk mod p) mod q s1=k-1(H(M1)+xr) mod q and s2=k-1(H(M2)+xr) mod q

We have s1 - s2 =k-1(H(M1)-H(M2)) mod q

Another attack (for OpenPGP) Replace p and g http://www.tigertools.net/board/?topic=topic4&msg=14 http://www.orlingrabbe.com/DSAflaw_OpenPGP.htm


Cont.

We cannot use small k


Non-deterministic

Non-determined signatures For each message, many valid signatures exist DSA, Elgamal

Deterministic signatures For each message, one valid signature exists RSA


Comparisons

Speed DSS has faster signing than verifying RSA could have faster verifying than signing Message be signed once, but verified many times

This prefers the faster verification But the signer may have limited computing power

Example: smart card This prefers the faster siging


Blind Signature (digital cash) first introduced by Chaum, allow a person to get a

message signed by another party without revealing any information about the message to the other party.

Suppose Alice has a message m that she wishes to have signed by Bob, and she does not want Bob to learn anything about m. Let (n,e) be Bob's public key and (n,d) be his private key. Alice generates a random value r such that gcd(r, n) = 1 and sends x = (re m)

mod n to Bob. The value x is ``blinded'' by the random value r; hence Bob can derive no useful information from it.

Bob returns the signed value t = xd mod n to Alice. Since xd (re m)d r md mod n, Alice can obtain the true signature s of m by computing s = r-1 t mod n.


Security Concerns

GnuPG permits creating ElGamal keys are usable for both encryption and signing. It is even possible to have one key (the primary

one) used for both operations. This is not considered good cryptographic practice,

but is permitted by the OpenPGP standard. signature is much larger than a RSA or DSA

signature verification and creation takes far longer and the

use of ElGamal for signing has always been problematic due to a couple of cryptographic weaknesses when not done properly.


Applications of Blind Signature

In an online context the blind signature works as follows. Voters encrypt their ballot with a secret key and then blinds it. Then the voter signs the encrypted vote and sends it to the

validator. The validator checks to see if the signature is valid (the signature

acts as a I.D. tag and will have to be registered with the voter before the voting process has started) and if it is the validator signs it and returns it to the voter.

The voter removes the blinding encryption layer, which then leaves behind an encrypted ballot with the validator's signature.


Cont.

This is then sent to the tallier who checks to make sure the validator's signature is present on the votes.

He then waits until all votes haven been collected and then publishes all the encrypted votes so that the voters can verify their votes have been received.

The voters then send their keys to the tallier to decrypt their ballots.

Once the vote has been counted the tallier publishes the encrypted votes and the decryption keys so that voters can then verify the results.

Next we illustrate the transfer of ballots between the various parties.


Cont.


Cont,

This protocol has been implemented used in reality and has been found that the entire voting process can be completed in a matter of minutes despite the complex nature of the voting procedure.

Most of the tasks can be automated with the only user interaction needed being the actual vote casting.

Encryption, blinding and all the verification needed can be performed by software in the background.

Of course we'd have to trust this software to handle the voting procedures correctly and accurately and to assume it has not been compromised in some way.



CertificateXiang-Yang Li


Certificate

A public-key certificate is a digitally signed statement from one entity, saying that the public key (and some other information) of another entity has some specific value.


More terms

Digitally Signed If some data is digitally signed it has been stored

with the "identity" of an entity, and a signature that proves that entity knows about the data. The data is rendered unforgeable by signing with the entitys' private key.

Identity A known way of addressing an entity. In some

systems the identity is the public key, in others it can be anything from a Unix UID to an Email address to an X.509 Distinguished Name.

Entity An entity is a person, organization, program,

computer, business, bank, or something else you are trusting to some degree.


More about CA

Why need it In a large-scale networked environment it is impossible to

guarantee that prior relationships between communicating entities have been established or that a trusted repository exists with all used public keys. Certificates were invented as a solution to this public key distribution problem. Now a Certification Authority (CA) can act as a Trusted Third Party. CAs are entities (e.g., businesses) that are trusted to sign (issue) certificates for other entities. It is assumed that CAs will only create valid and reliable certificates as they are bound by legal agreements. There are many public Certification Authorities, such as VeriSign, Thawte, Entrust, and so on. You can also run your own Certification Authority using products such as the Netscape/Microsoft Certificate Servers or the Entrust CA product for your organization.

http://www.verisign.com/

http://www.thawte.com/

http://www.entrust.com/


Who uses Certificate?

Probably the most widely visible application of X.509 certificates today is in web browsers (such as Netscape Navigator and Microsoft Internet Explorer) that support the SSL protocol. SSL (Secure Socket Layer) is a security protocol that provides

privacy and authentication for your network traffic. These browsers can only use this protocol with web servers that support SSL.

Other technologies that rely on X.509 certificates include: Various code-signing schemes, such as signed Java Archives, and

Microsoft Authenticode. Various secure E-Mail standards, such as PEM and S/MIME. E-Commerce protocols, such as SET.


How to create certificate?

There are two basic techniques used to get certificates: you can create one yourself (using the right tools, such as keytool)

Not everyone will accept self-signed certificates, you can ask a Certification Authority to issue you one (either directly or

using a tool such as keytool to generate the request).

The main inputs to the certificate creation are: Matched public and private keys, generated using some special tools

(such as keytool), or a browser. information about the entity being certified (e.g., you). This normally

includes information such as your name and organizational address. If you ask a CA to issue a certificate for you, you will normally need to provide proof to show correctness of the information.

http://csrc.nist.gov/wireless/S10_802.11i%20Overview-jw1.pdf


business

Many companies sale the service of creating the certificate (such as SSL certificate) Comodo Verisign Thawte Entrust Geotrust


X.509 Authentication Service

Public key certificate associated with user The certificates are created by Trusted Authority Then placed in the directory by TA or user Itself is not responsible for creating certificate It includes

Version, serial number, signature algorithm identifier, Issuer name, issuer identifier, validity period, the user, user identifier, user’s public key, extensions, signature by TA

The signature by TA guarantees the authority Certificates can be used to certify other TAs Y<<X>>: certificate of user X issued by TA Y


What is inside X.509 certificate? Version

Thus far, three versions are defined. Serial Number

distinguish it from other certificates it issues. This information is used in numerous ways, for example when a certificate is revoked its serial number is placed in a Certificate Revocation List (CRL).

Signature Algorithm Identifier This identifies the algorithm used by the CA to sign the

certificate. Issuer Name

The X.500 name of the entity that signed the certificate. This is normally a CA. Using this certificate implies trusting the entity that signed this certificate. root or top-level CA certificates, the issuer signs its own certificate.


cont Validity Period

This period is described by a start date and time and an end date and time, and can be as short as a few seconds or almost as long as a century. It depends on a number of factors, such as the strength of the private key used to sign the certificate or the amount one is willing to pay for a certificate. This is the expected period that entities can rely on the public value, if the associated private key has not been compromised.

Subject Name The name of the entity whose public key the certificate

identifies. This name uses the X.500 standard, so it is intended to be unique across the Internet.

Subject Public Key Information together with an algorithm identifier


Certificate Revocation

Need the private key together with the certificate to revoke it

The revocation is recorded at the directory

Each time a certificate is arrived, check the directory to see if it is revoked


X.509 Authentication Service

part of CCITT X.500 directory service standards distributed servers maintaining some info database

defines framework for authentication services directory may store public-key certificates with public key of user signed by certification authority

also defines authentication protocols uses public-key crypto & digital signatures

algorithms not standardised, but RSA recommended


X.509 Certificates

issued by a Certification Authority (CA), containing: version (1, 2, or 3) serial number (unique within CA) identifying certificate signature algorithm identifier issuer X.500 name (CA) period of validity (from - to dates) subject X.500 name (name of owner) subject public-key info (algorithm, parameters, key) issuer unique identifier (v2+) subject unique identifier (v2+) extension fields (v3) signature (of hash of all fields in certificate)

notation CA<<A>> denotes certificate for A signed by CA


X.509 Certificates


Obtaining a Certificate

any user with access to CA can get any certificate from it

only the CA can modify a certificate because cannot be forged, certificates

can be placed in a public directory


CA Hierarchy

if both users share a common CA then they are assumed to know its public key

otherwise CA's must form a hierarchy use certificates linking members of

hierarchy to validate other CA's each CA has certificates for clients (forward) and

parent (backward) each client trusts parents certificates enable verification of any certificate

from one CA by users of all other CAs in hierarchy


CA Hierarchy Use


Certificate Revocation

certificates have a period of validity may need to revoke before expiry, eg:

1. user's private key is compromised

2. user is no longer certified by this CA

3. CA's certificate is compromised

CA’s maintain list of revoked certificates

the Certificate Revocation List (CRL)

users should check certs with CA’s CRL


Authentication Procedures

X.509 includes three alternative authentication procedures:

One-Way Authentication Two-Way Authentication Three-Way Authentication all use public-key signatures


One-Way Authentication

1 message ( A->B) used to establish the identity of A and that message is from A message was intended for B integrity & originality of message

message must include timestamp, nonce, B's identity and is signed by A


Two-Way Authentication

2 messages (A->B, B->A) which also establishes in addition: the identity of B and that reply is from B that reply is intended for A integrity & originality of reply

reply includes original nonce from A, also timestamp and nonce from B


Three-Way Authentication

3 messages (A->B, B->A, A->B) which enables above authentication without synchronized clocks

has reply from A back to B containing signed copy of nonce from B

means that timestamps need not be checked or relied upon


X.509 Version 3

has been recognised that additional information is needed in a certificate email/URL, policy details, usage constraints

rather than explicitly naming new fields defined a general extension method

extensions consist of: extension identifier criticality indicator extension value


Certificate Extensions

key and policy information convey info about subject & issuer keys, plus indicators

of certificate policy

certificate subject and issuer attributes support alternative names, in alternative formats for

certificate subject and/or issuer

certificate path constraints allow constraints on use of certificates by other CA’s



IdentificationXiang-Yang Li


Identification

Identification: user authentication convince system of your identity before it can act on your behalf sometimes also require that the computer verify its identity with

the user

Based on three methods what you know what you have what you are

Verification Validation of information supplied against a table of possible

values based on users claimed identity


What you Know

Passwords or Pass-phrases prompt user for a login name and password verify identity by checking that password is correct on some (older) systems, password was stored clear more often use a one-way function, whose output

cannot easily be used to find the input value either takes a fixed sized input (eg 8 chars) or based on a hash function to accept a variable sized

input to create the value important that passwords are selected with care to

reduce risk of exhaustive search


Weakness

Traditional password scheme is vulnerable to eavesdropping over an insecure network


Solutions?

One-time password these are passwords used once only future values cannot be predicted from older values

Password generation either generate a printed list, and keep matching list on

system to be accessed or use an algorithm based on a one-way function f (eg

MD5) to generate previous values in series (eg SKey) start with a secret password s, and number N , p0

= fN(s) ith password in series is pi = fN-i(s)

must reset password after N uses


What you Have

Magnetic Card, Magnetic Key possess item with required code value encoded

Smart Card or Calculator may interact with system may require information from user could be used to actively calculate: a time dependent password a one-shot password a challenge-response verification public-key based verification


What you Are

Verify identity based on your physical characteristics, known as biometrics

Characteristics used include: Signature (usually dynamic) Fingerprint, hand geometry face or body profile Speech, retina pattern

Tradeoff between false rejection (type I error) false acceptance (type II error)



Secret SharingXiang-Yang Li


Threshold Scheme

A (t,w)-threshold scheme Sharing key K among a set of w users Any t users can recover the key Any t-1 users can not do so

Schemes Shamir’s scheme Geometric techniques Matroid theory


Information Theory

The secret sharing is as large as the original secret This result is based in information theory, but can be understood

intuitively. Given t-1 shares, no information whatsoever can be determined about the secret. Thus, the final share must contain as much information as the secret itself.

All secret sharing schemes use random bits. To distribute a one-bit secret among threshold t people, t-1 random

bits are necessary. The final share contains as much information as the secret, but the other t-1 shares still provide relevant information individually. This information cannot be the secret, so it must be random.

http://en.wikipedia.org/wiki/Information_theory

http://en.wikipedia.org/wiki/Random



Shamir’s Scheme

Initialization phase Dealer chooses a large prime number p Dealer chooses w distinct xi from Zp

Gives value xi to person pi

Share distribution of key k from Zp

Dealer choose t-1 random number ai

Dealer computes yi=f(xi) Here f(x)=k+ajxj mod p

Dealer gives share yi to person pi


Geometry View

http://www.rsasecurity.com/rsalabs/faq/images/shamir.gif


Simple (t,t) Sharing

Procedure D secretly chooses t-1 random elements yi from Zn

D computes Value yt=K- yj mod n

D distributes yi to person pi for all i

It is secure and easy Number n can be any number Easy to recover the key Only t persons together can do so, assume yi random


Blakley's Scheme

Secret is a point in an t-dimensional space

Dealer gives each user a hyper-plane passing the secret point

Any t users can recover the common point


Geometry View

http://www.rsasecurity.com/rsalabs/faq/images/blakley.gif


Avoid Cheating

Two major distinct weaknesses Bogus values are undetectable. Participants need not reveal their true share.

Even if a bogus value was detected, it would not necessarily give any information about the true value

One participant did not reveal its true value after get the true values from other one


Ben-Or/Rabin Solution

Using Checking Vectors For any two participants A and B

Dealer gives A (SA, YAB)

Dealer gives B (BAB, CAB)

Here CAB = BAB YAB+ SA mod p

SA is the secret share of A

A and B keep their values secret B can use (BAB, CAB) to verify the value (SA, YAB) of A


Avoid Cheating

Participant B can send A bogus value after receive A’s value

Solution: bit transfer Dealer gives A (SAi, YABi)

Dealer gives B (BABi, CABi)

Here CABi = BABi YABi+ SAi mod p

SAi is the ith bit of the secret share of A


Cont.

Protocol Participant A gives its value (SAi, YABi) to B

B verifies: CABi = BABi YABi+ SAi mod p

B then sends its value (SBi, YBAi) to A

A verifies: CBAi = BBAi YBAi+ SBi mod p

The protocol terminates whenever One side detects cheating, or All values transferred


Chinese Remainder Theorem

Given a number m<n, and n=n1n2…nk, Numbers ni and nj are coprimes

Let ai=m mod ni

Number n is public Dealer delivers ai and ni to the ith participant

Then all k users can recover the number m

Why it is not a good secret sharing scheme? Is it computationally for any k-1 users to recover the

key if n is large?


Recover method

Each user pre-computes Ni=n/ni

Inverse of Ni: yi=Ni mod ni

Compute the product si=aiNiyi mod n

Recover the secret m Each user submits si

Computes s1+s2+….+sk mod n


Access Structure

Threshold scheme allows any t users to recover key!

Access structure allows some subsets to recover the key! Example: {{p1,p2,p4},{p1,p3,p4},{p2,p3}} among

p1,p2,p3,p4,p5 able to recover the key

Assume the accessing subset is minimized No subset of any accessing subset is able to

recover


Monotone Circuit

Assign sharing for each accessing subset

k

kkk

p1 p2 p3 p4

a1a2

b1

b2 k-b1-b2

k-a1-a2

c1 k-c1


Cont.

Distribution (a1,b1) to p1

(a2,c1) to p2

(k-c1,b2) to p3

(k-a1-a2,k-b1-b2) to p4

The sharer needs know The circuit used by dealer Which shares corresponding to which wires

The shared value is secret


Visual Secret Sharing

There is a secret picture to be shared among n participants. The picture is divided into n transparencies (shares)

such that if any m transparencies are placed together, the picture

becomes visible but if fewer than m transparencies are placed together,

nothing can be seen.


Visual Secret Sharing Such a scheme is constructed by

viewing the secret picture as a set of black and white pixels and handling each pixel separately. The schemes are perfectly secure and easily

implemented without any cryptographic computation. A further improvement allows each

transparency (share) to be an innocent picture For example, a picture of a landscape or a picture of a

building thus concealing the fact of secret sharing


Interactive Proof

Interactive proof is a protocol between two parties in which one party, called the prover, tries to prove a certain fact to the other party, called the verifier

Often takes the form of a challenge-response protocol


cont

protocol in which one or more provers try to convince another party, called the verifier, that the prover(s) possess certain true knowledge, such as the membership of a string x in a given language, often with the goal of revealing no further details about this knowledge. The prover(s) and verifier are formally defined as probabilistic Turing machines with special "interaction tapes" for exchanging messages.

http://www.nist.gov/dads/HTML/string.html

http://www.nist.gov/dads/HTML/language.html

http://www.nist.gov/dads/HTML/probablturng.html


Desired Properties

Desired properties of interactive proofs Completeness: The verifier always accepts the proof if

the prover knows the fact and both the prover and the verifier follow the protocol.

Soundness: Verifier always rejects the proof if prover doesnot know the fact, and verifier follows protocol.

Zero knowledge: The verifier learns nothing about the fact being proved (except that it is correct) from the prover that he could not already learn without the prover. In a zero-knowledge proof, the verifier cannot even later prove the fact to anyone else.


Typical Protocol

A typical round in a zero-knowledge proof consists of a "commitment" message from the prover, followed by a challenge from the verifier, and then a response to the challenge from the prover. The protocol may be repeated for many rounds. Based on the prover's responses in all the rounds, the verifier decides whether to accept or reject the proof.


An example

Ali Baba’s Cave


Cont.

Alice wants to prove to Bob that she knows the secret words to open the portal at CD but does not wish to reveal the secret to Bob. In this scenario, Alice’s commitment is to go to C or D.


Proof Protocol

A typical round in the proof proceeds as follows: Bob goes to A, waits there while Alice goes to C or D. Bob then asks Alice to appear from either the right side

or the left side of the tunnel. If Alice does not know the secret words

there is only a 50 percent chance that she will come out from the right tunnel.

Bob will repeat this round as many times as he desires until he is certain that Alice knows the secret words.

No matter how many times that the proof repeats, Bob does not learn the secret words.


Graph Isomorphism

Problem Instance Two graphs G1=(V1,E1) and G2=(V2,E2)

Question Is there a bijection f from V1 to V2, so (u,v)E1 implies

that (f(u),f(v))E2

If such bijection exists, then graphs G1 and G2 are said to be isomorphic

If such bijection does not exist, then graphs G1 and G2 are said to be non-isomorphic


Graph Non-isomorphism

Input: graphs G1 and G2 over {1,2,…n} Prover want to prove

G1 and G2 are not isomophic

Assumption Prover has unbounded computational power Verifier has limited computational power


Proof Protocol

Protocol (repeated for n rounds) Verifier

Randomly chooses i=1 or 2 Selects a random permutation f and compute H to

be the image of Gi under f, sends H to prover

Prover Determines the value j such that Gj is isomorphic

to H Sends j to verifier

Verifier checks if j=i If equal for n rounds, then accepts the proof


Correctness and Soundness

Correctness If G1 and G2 are not isomorphic, then for any round,

there is only one graph of G1, G2 that could produce H under a permutation f

So if the verifier knows non-isomorphism, then each round a correct j will be computed

Soundness If the verifier does not know (G1 and G2 are

isomorphic), then each round two answers possible, and it has half chance to get the correct i chosen by the prover.


Graph Isomorphism

Input: graphs G1 and G2 over {1,2,…n} Prover want to prove

G1 and G2 are isomophic

Assumption Prover has unbounded computational power Verifier has limited computational power


Proof Protocol

Protocol (repeated for n rounds) Prover

Selects a random permutation f and compute H to be the image of G1 under f, sends H to prover

Verifier Randomly chooses i=1 or 2, sends it to prover

Prover Computes the permutation g such that H is the image

of Gj under g, and sends g to verifier Verifier

checks if H is the image of Gj under g If yes for n rounds, then accepts the proof


Correctness and Soundness

Correctness If G1 and G2 are isomorphic, and the verifier knows

how to find the permutation between G1 and G2, then each round a correct g will be computed

Soundness If the verifier does not know (G1 and G2 are non-

isomorphic or the permutation between G1 and G2), then each round prover can deceive the verifier is to guess the value i chosen by the verifier


Perfect Zero-Knowledge

The graph isomorphism proof is ZKP All information seen by the verifier is the same as

generated by a random simulator Define transcript of the proof as

t=(G1,G2,(H1,i,g1),(H2,i,g2),….(Hn,i,gn))

Anyone can generate the transcript without knowing which permutation carries G1 to G2

Hence the verifier gains nothing by knowing the transcript (I.e., the proof history)


ZKP for Verifier

Perfect Zero-knowledge for verifier Suppose we have a poly-time interactive proof system

and a poly-time simulator S. Let T be all yes-instance transcripts and let F be all transcripts generated by S. For any transcript t if Pr(t occurs in T)=Pr(t occurs in F)

We say the interactive proof system are perfect zero-knowledge for the verifier


Isomorphism Proof: ZKP-verifier

Graph isomorphism is a perfect zero-knowledge for verifier A triple (H,i,g). There are 2n! valid triples. All triples (H,i,g) occurs equiprobable in some

transcript Here, assume that both the verifier and the

prover are honest Both of them randomly chooses parameters that

supposed to be chosen randomly


Cheating Verifier

What happened if verifier does not follow the protocol (does not choose i randomly) Transcript produced by ZKP is not same as that

produced by the random simulator anymore The verifier may gain some information due to this

imbalance But, there is another expected poly-time simulator to

generate the same transcript Hence, the verifier still gains nothing


Perfect Zero-Knowledge

Definition Suppose we have a poly-time interactive proof system,

a poly-time algorithm V to generate random numbers by verifier, and a poly-time simulator S. Let T be all yes-instance transcripts (depending on V) and let F be all transcripts generated by S and V. For any transcript t if Pr(t occurs in T)=Pr(t occurs in F)

We say the interactive proof system are perfect zero-knowledge


Forging Simulator

Initial transcript t=(G1,G2), repeat n rounds Let old-state=state(V), repeat follows

Chooses ij from {1,2} randomly

Chooses gj to be a random permutation over {1,...n}

Compute Hj to be the image of Gi under g

Call V with input Hj, obtaining a challenge ij’

If ij=ij’, then concatenate (Hj, ij, gj) onto the end of t

Else reset V by state(V)=old-state

Until ij=ij’


Perfect Zero-knowledge

The graph isomorphism is perfect ZKP The expected running time of simulator is 2n For the kth round of the interactive proof system

Let pk be the probability that verifier chooses i=1

Then (H,1,g) occurs in actual transcript with pk/n!, (H,2,g) occurs in actual transcript with (1-pk)/n!

For simulator, when it terminates the simulation for the kth round, same probability distribution for (H,1,g) and (H,2,g)

Therefore, all transcripts by simulator or actual has the same probability distribution


Quadratic Residue

Fiat-Shamir Identification Question

Given integer n=pq, here p, q are primes. Prover wants to prove

Integer x is a quadratic residue mod n In other words, knows u so x=u2 mod n

Quadratic residue is hard to solve if do not knowing the factoring of n


Proof Protocol

Repeat the following for log2n times Prover

Chooses random v less than n and computes y=v2 mod n. Sends y to verifier

Verifier Chooses a random I from {0,1}, sends it to prover

Prover Computes z=u2v mod n, sends z to verifier

Verifier Checks if z2=xiy mod n

Accepts the proof if equation holds all log2n rounds


Cont

Correctness Show that verifier will accept the prover if indeed

knows

Soundness Show that verifier will detect the prover if it does not

know with a good probability

Zero-knowledge Show that verifier gets nothing from the protocol


Guillou Quisquater Protocol

The GQ protocol is an extension of the Fiat Shamir protocol that limits the number t of rounds required.

One Time Set-up: 1. A trusted authority T selects two random

primes p and q and forms a modulus n = p · q.

2. T defines a public exponent v > 4 with gcd(v, (p-1)(q -1) = 1 so that T can compute s = v-1 mod (p-1) (q-1).

3. T publishes parameters n and v.


Cont.

Selection of per-user parameters: 1. Each entity A has a unique identification

Id(A). Everyone can calculate a value J(A) = f(Id(A)) mod n (the redundant identity).

2. T gives to each entity A the secret data secret(A) = J(A)-s, which it can calculate.


Cont. Protocol: A proves her identity to B using t rounds, each of

which consists of: 1. A selects a random secret r and sends her identity Id(A)

and x = rv mod n to B. 2. B selects a random challenge e in {1, 2, ... , v}. 3. A computes and sends the following response to B: y = r ·

secret(A)e mod n. 4. B receives y, constructs J(A) = f(Id(A)) mod n, computes z

= J(A)eyv, and accepts this round if z = x mod n. In this protocol, v determines the security level. In Fiat

Shamir, v = 2 and there are many rounds. A fraudulent claimant can defeat the protocol by correctly guessing the challenge e (with a 1 in v chance.) GQ seems secure, because we need to extract v-roots modulo n.


Discrete Logarithm

Question: Prover wants to prove to verifier that he knows x such

that y=gx mod p . Here g, y, and p are public information Prover does not want to publicize the value of x.


Proof Protocol

Repeat the following for log2n times Prover

Chooses random j < p-1 and computes r=gj mod p. Sends r to verifier

Verifier Chooses a random i from {0,1}, sends it to prover

Prover Computes h=i x +j mod p-1, sends h to verifier

Verifier Checks if gh=yir mod n

Accepts the proof if equation holds all log2n rounds


Cont

Correctness Show that verifier will accept the prover if indeed

knows

Soundness Show that verifier will detect the prover if it does not

know with a good probability

Zero-knowledge Show that verifier gets nothing from the protocol


Bit Commitments

Bit commitment Sometimes, it is desirable to give someone a piece of

information, but not commit to it until a later date. It may be desirable for the piece of information to be held secret for a certain period of time.

Example: stock up and down


Properties

Bit commitment scheme The sender encrypts the b in some way The encrypted form of b is called blob Scheme f: (X,b)Y

Properties Concealing: verifier cannot detect b from f(x,b) Binding: sender can open the blob by revealing x Hence, the sender must use random x to mask b


Methods

One can choose any encryption method E Function f((x0,k),b)=Ek((x0,b))

Need supply decryption k to reveal b Assume the decryption method D is known

Choose any integer n=pq, p and q are large primes Function f(x,b)=mbx2 mod n

Goldwasser-Micali Scheme Here n=pq, m is not quadratic residule, m,n public mx1

2 mod n x22 mod n

So sender can not change mind after commitment


Coin Flip

Even protocols Alice has a coin flip result i or j Bob wants to guess the result Alice has a message M that is commitment If bob guesses correct, Bob should have M received Alice starts with 2 pairs of public keys (Ei,Di) and

(Ej,Dj) Bob starts with a symmetric encryption S and a key k


Protocol

Procedure Alice sends Ei, Ej to Bob Bob guess h and sends y=Eh(k) to Alice Alice computes p=Dj(y) and sends the encryption z of

M by p using S to Bob Bob decrypts the encryption z using S and key k If the guess is correct, then Bob gets the commitment


Oblivious Transfer

What is oblivious transfer Alice wants to send Bob a secret in such a way that Bob

will know whether he gets it, but Alice won't. Another version is where Alice has several secrets and transfers one of them to Bob in such a way that Bob knows what he got, but Alice doesn't. This kind of transfer is said to be oblivious (to Alice).


Transfer Factoring

By means of RSA, oblivious transfer of any secret amounts to oblivious transfer of the factorization of n=pq Bob chooses x and sends x2 mod n to Alice Alice (who knows p,q) computes the square roots x,-

x,y,-y of x2 mod n and sends one of them to Bob. Note that Alice does not know x.

If Bob gets one of y or -y, he can factor n. This means that with probability 1/2, Bob gets the secret. Alice doesn't know whether Bob got one of y or -y because she doesn't know x.


Factoring

If one knows x and y such that 1) x2=y2 mod n 2) 0<x,y<n, xy and x+y0 mod n Number n is the production of two primes

Then n can be factored First gcd(x+y,n) is a factor of n And gcd(x-y,n) is a factor of n


Quadratic Solution

Given n=p, and a is a quadratic residue Then there is two positive integers x less than n Such that x2=a mod n

Given n=pq, and a is a quadratic residue Then there is four positive integers x less than n Such that x2=a mod n


Oblivious Transfer of Message

Alice has a message M, bob wants to get M through oblivious transfer Alice does not know if Bob get M or not Bob knows if he gets it or not Bob gets M with probability ½ Coin flipping can be used to achieve this


Contract Signing

It requires two things Commitment: after certain point, both parties are bound

by the contract, until then, neither is Unforgeability: it must be possible for either party to

prove the signature of the other party

With Pen and Paper Two party together, face to face Sign simultaneously (or one character by one)


Remote Contract Signing

Simple one Alice generate a signature, divided into SL, SR Alice randomly select two keys KL, KR Encrypt the signatures SL, SR Transfer encrypted SL,SR to Bob Obliviously transfer KL, KR to bob

Bob gets one, but Alice does not know which one Bob decrypts the encrypted SL or SR

Verify the decrypted signature, if invalid, stop Alice sends the ith bits of keys KL and KR to Bob

Here i=1 to the length of the keys


Cont.

The protocol will be conducted by Bob also What is the chance of Alice to cheat successfully?

Alice can guess which key will be transferred obliviously ---(1/2 chance)

Then send wrong signature for the other half or send the wrong key of the other half

Bob can not detect it if Alice can guess which key Bob got

How about Alice stop prematurely? One bit advance over Bob

Enhanced protocol Use many pair of keys and signatures instead of one



Pseudo-random NumberXiang-Yang Li


Random number, Pseudorandom The outputs of pseudorandom number

generators are not truly random they only approximate some of the properties of

random numbers. "Anyone who considers arithmetical methods of

producing random digits is, of course, in a state of sin.”--- John von Neumann

Truely random numbers can be generated using hardware random number generators


Randomness Definition

Chaitin-Kolmogorov randomness (also called algorithmic randomness) a string of bits is random if and only if it is shorter than

any computer program that can produce that string this basically means that random strings are

those that cannot be compressed.

Statistical Randomness A numeric sequence is said to be statistically random

when it contains no recognizable patterns or regularities; sequences such as the results of an ideal die roll,

or the digits of Pi (as far as we can tell) exhibit statistical randomness.

http://en.wikipedia.org/wiki/Chaitin-Kolmogorov_randomness

http://en.wikipedia.org/wiki/Chaitin-Kolmogorov_randomness


http://en.wikipedia.org/wiki/Data_compression

http://en.wikipedia.org/wiki/Dice

http://en.wikipedia.org/wiki/Pi


Inherent non-randomness

Because any PRNG run on a deterministic computer (contrast quantum computer) is deterministic, its output will inevitably have certain properties that a true random sequence would not exhibit. guaranteed periodicity—it is certain that if the generator uses only

a fixed amount of memory then, given a sufficient number of iterations, the generator will revisit the same internal state twice, after which it will repeat forever. A generator that isn't periodic can be designed, but its memory requirements would grow as it ran. In addition, a PRNG can be started from an arbitrary starting point, or seed state, and will always produce an identical sequence from that point on.


cont

In practice, many PRNGs exhibit artifacts which can cause them to fail statistically significant tests. These include, but are certainly not limited to: Shorter than expected periods for some seed

states (not full period) Poor dimensional distribution Successive values are not independent Some bits may be 'more random' than others Lack of uniformity


Pseudo-random Bit Generator

Several applications Key generation Some encryption algorithms, or one-time pad

Let l>k be integers Function f: Z2

k Z2l computable in poly-time

Then f called (k,l)-pseudo-random bit generator The input s0 Z2

k is called the seed

Output f(s0) is called the pseudo-random string


Desired Properties

Three important properties: Unbiased (uniform distribution):

All values of whatever sample size is collected are equiprobable

Unpredictable (independence): It is impossible to predict what the next output

will be, given all the previous outputs, but not the internal "hidden" state.

Irreproducible: Two of the same generators, given the same

starting conditions, will produce different outputs.


Desired Properties

Usually when a person says A "good" pseudo-random number generator

they mean it is unbiased. A "true" PRNG

they usually mean it's irreproducible A "cryptographically strong" PRNG

they mean it's unpredictable Very rarely they mean it's all threes


More Properties

Long period The generator should be of long period

Fast computation The generator should be reasonably fast

Security The generator should be secure What is security level of PRNG?


Security

A PRNG suitable for cryptographic applications is called a cryptographically secure PRNG (CSPRNG). Its output should not only pass all statistical tests for randomness but satisfy

some additional cryptographic requirements. Used in many aspects of cryptography require random numbers, for example:

Key generation Nonces Salts in certain signature schemes, (ECDSA, RSASSA-PSS). One-time pads


CSPRNG CSPRNG requirements fall into two groups:

their statistical properties are good (passing tests of randomness), they hold up well in case of attack, even when (part of) their secrets are

revealed. A CSPRNG should satisfy the 'next-bit test'.

Given the first l bits of a random sequence there is no polynomial-time algorithm that can predict the next bit with probability of success significantly higher than 1/2.

It has been proven that a generator passing the next-bit test will pass all other polynomial-time statistical tests for randomness.

should withstand state compromise extensions. That is, in the unfortunate case that part or all of the state has been revealed

(or guessed correctly), it should be impossible to reconstruct the stream of random numbers prior to the incident. Also if there is an input of entropy, it should be infeasible to use knowledge of the state to predict future conditions of the state.


Example

the CSPRNG being considered produces output by computing some function of the next digit of pi (ie, 3.1415...),

it may well be random as pi appears to be a random sequence.

However, this does not satisfy the next-bit test, and thus is not cryptographically secure. There exists an algorithm that will predict the next bit.


Design

divide designs of CSPRNGs into classes: those based on block ciphers; those based upon hard mathematical problems, and special-purpose designs.


Designs based on cryptographic primitives Designs based on cryptographic primitives

A secure block cipher can also be converted into a CSPRNG by running it in counter mode.

This is done by choosing an arbitrary key and encrypting a zero, then encrypting a 1, then encrypting a 2, etc. The counter can also be started at an arbitrary number other than zero. Obviously, the period will be 2n for an n-bit block cipher; equally obviously, the initial values (i.e. key and 'plaintext') must not become known to an attacker lest, however good this CSPRNG construction might be otherwise, all security be lost.

A cryptographically secure hash of a counter might also act as a good CSPRNG in some cases. it is necessary that the initial value of this counter is random and secret.

If the counter is a bignum, then CSPRNG could have an infinite period.


DES Based Generator

ANSI X9.17 PRNG (used by PGP,..) Inputs: two pseudo-random inputs

one is a 64-bit representation of date and time The other is 64-bit seed values

Keys: three 3DES encryptions using same keys Output:

a 64-bit pseudorandom number and A 64-bit seed value for next-round use


ANSI X9.17

EDE

EDE

EDE

DT

Si

Ri

Si+1

K1,K2


Linear Congruential Generator

Protocol Let M be an integer and a, b less than M Let k be number of bits of M Integer l is between k+1 and M-1 Let s0 be a seed less than M

Define si=asi-1+b mod M

Then the ith random bit is si mod 2

It is not proved to be secure


Parameter Setting

Not all a, b are good and m should be large

For example, m is a large prime number For fast computation, usually m=231-1

And b is set to 0 often

For this m, there are less than 100 integers a It generates all numbers less than m The generated sequences appear to be random

One such a=7516807

Used in IBM 360 family of computers


RSA Generator

Protocol Let p, q be two k/2 bits primes and define n=pq Integer b: gcd(b, (n))=1 Public: n, b; Private p,q A seed s0 with k bits

Sequence si+1=sib mod n

Then the ith random bit is si mod 2

It is proved to be secure!


BBS Generator

Blum-Blum-Shub Generator Let p, q be two k/2 bits primes and define n=pq Here p=q=3 mod 4

this guarantees that each quadratic residue has one square root which is also a quadratic residue

gcd(φ(p-1), φ(q-1)) should be small this makes the cycle length large.

Let QR(n) be all quadratic residues modulo n Public: n; Private p,q A seed s0 with k bits from QR(n) Sequence si+1=si

2 mod n Then the ith random bit is si mod 2


Cont on BBS

Provably “secure” When the primes are chosen appropriately, and O(log log n) bits of each Si are output,

then in the limit as n grows large, distinguishing the output bits from random will be at least as difficult as factoring n.

However, it's theoretically possible that a fast algorithm for

factoring will someday be found, so BBS is not yet guaranteed to be secure.


Discrete Logarithm Generator

Protocol Let p be a k-bit prime, Let be primitive element modulo p A seed s0 is any non-zero integer less than p

Define si+1 = si mod p

Then the ith random bit is 1 if si is larger than p/2

0 if si is less than p/2


Standards

A number of designs of CSPRNGs have been standardized. They can be found in: FIPS 186-2 ANSI X9.17-1985 Appendix C ANSI X9.31-1998 Appendix A.2.4 ANSI X9.62-1998 Annex A.4


Network Security


Topics to be covered

Applications Email security www security Malicious software

Networks Wireless LAN security 802.11 IPsec Firewall Intrusions



Email SecurityXiang-Yang Li


Electronic Mail Security

Despite the refusal of VADM Poindexter and LtCol North to appear, the Board's access to other sources of information filled much of this gap. The FBI provided documents taken from the files of the National Security Advisor and relevant NSC staff members, including messages from the PROF system between VADM Poindexter and LtCol North. The PROF messages were conversations by computer, written at the time events occurred and presumed by the writers to be protected from disclosure. In this sense, they provide a first-hand, contemporaneous account of events.—The Tower Commission Report to President Reagan on the Iran-Contra Affair, 1987


Email Security

email is one of the most widely used and regarded network services

currently message contents are not secure may be inspected either in transit or by suitably privileged users on destination system


Email Security Enhancements

confidentiality protection from disclosure

authentication of sender of message

message integrity protection from modification

non-repudiation of origin protection from denial by sender


Pretty Good Privacy (PGP)

widely used de facto secure email developed by Phil Zimmermann selected best available crypto algs to

use integrated into a single program available on Unix, PC, Macintosh and

Amiga systems originally free, now have commercial

versions available also


PGP

Five services Authentication, confidentiality, compression, email

compatibility, segmentation

Functions Digital signature Message encryption Compression Email compatibility segmentation


PGP Operation – Authentication

1. sender creates a message2. SHA-1 used to generate 160-bit hash code of

message3. hash code is encrypted with RSA using the

sender's private key, and result is attached to message

4. receiver uses RSA or DSS with sender's public key to decrypt and recover hash code

5. receiver generates new hash code for message and compares with decrypted hash code, if match, message is accepted as authentic


PGP Operation – Confidentiality

1. sender generates message and random 128-bit number to be used as session key for this message only

2. message is encrypted, using CAST-128 / IDEA/3DES with session key

3. session key is encrypted using RSA with recipient's public key, then attached to message

4. receiver uses RSA with its private key to decrypt and recover session key

5. session key is used to decrypt message


PGP Operation – Confidentiality & Authentication

uses both services on same message create signature & attach to message encrypt both message & signature attach RSA encrypted session key


PGP Operation – Compression

by default PGP compresses message after signing but before encrypting so can store uncompressed message & signature for

later verification & because compression is non deterministic

uses ZIP compression algorithm


PGP Operation – Email Compatibility

when using PGP will have binary data to send (encrypted message etc)

however email was designed only for text

hence PGP must encode raw binary data into printable ASCII characters

uses radix-64 algorithm maps 3 bytes to 4 printable chars also appends a CRC

PGP also segments messages if too big


PGP Operation – Summary


Segmentation & Reassembly

Email systems impose maximum length 50 Kb, for example

PGP provides automatic segmentation Done after all other operations Thus only one session key needed


Key management

Generating unpredictable session keys Identifying keys

Multiple public, private key pairs for a user

Maintain keys Its own public, private keys of a PGP entity Public keys of correspondents


Session Key Generation

Algorithm used: CAST-128 Input to CAST-128

A 128-bit key Two 64 bits plaintexts to be encrypted

Output using cipher feedback mode Generates 2 64-bits ciphers form session key

Plaintexts are from 128-bits randomized number Based on key stroke of user (timing and actual keys) Then combined with previous session key


Key Identifiers

Receiver has multiple public keys How to know which private key is proper?

Approach Sending the least significant 64 bits as key ID Need send the receiver’s public key ID used for

encrypting the session key Need send the sender’s public key ID, whose

corresponding private key used for signature


Key Rings

Private key rings Timestamp, Key ID, public key, encrypted private key,

user ID

Public key rings Timestamp, Key ID, public key, owner trust, user ID,

key legitimacy, signature, signature trust


Public Key Management

A public key attributed to B may belong to C C can send messages to A forge B’s sigC can read any encrypted message to B

Approach to true public key Physically get key from B Obtain B’s key from mutual trusted authority Using key legitimacy field

computed from the signature trust field and number of certificates for the key


Revoking Public Key

Reason It is compromised: private key is open Simply to avoid use of same key for a period

Approach Owner issues key revocation certificate, signed by

owner Using corresponding private key to sign the certificate Disseminate the certificate as widely and as quickly as

possible


S/MIME (Secure/Multipurpose Internet Mail Extensions)

security enhancement to MIME email original Internet RFC822 email was text only MIME provided support for varying content types and

multi-part messages with encoding of binary data to textual form S/MIME added security enhancements

have S/MIME support in various modern mail agents: MS Outlook, Netscape etc


S/MIME Functions

enveloped data encrypted content and associated keys

signed data encoded message + signed digest

clear-signed data cleartext message + encoded signed digest

signed & enveloped data nesting of signed & encrypted entities


S/MIME Cryptographic Algorithms

hash functions: SHA-1 & MD5 digital signatures: DSS & RSA session key encryption: ElGamal & RSA message encryption: Triple-DES, RC2/40

and others have a procedure to decide which

algorithms to use


S/MIME Certificate Processing

S/MIME uses X.509 v3 certificates managed using a hybrid of a strict

X.509 CA hierarchy & PGP’s web of trust each client has a list of trusted CA’s

certs and own public/private key pairs & certs certificates must be signed by trusted

CA’s


Certificate Authorities

have several well-known CA’s Verisign one of most widely used Verisign issues several types of Digital IDs with increasing levels of checks & hence

trustClass Identity Checks Usage

1 name/email check web browsing/email

2+ enroll/addr check email, subs, s/w validate

3+ ID documents e-banking/service access


Email SPAM

Spam is flooding the Internet with many copies of the same message, in an attempt to force the message on people who would not otherwise choose to receive it. Most spam is commercial advertising, often for dubious products, get-rich-quick schemes, or quasi-legal services. Spam costs the sender very little to send -- most of the costs are paid for by the recipient or the carriers rather than by the sender


Email Spam

E-mail spam has existed since the beginning of the Internet, and has grown to about 90 billion messages a day, although about 80% is sent by fewer than 200 spammers. Botnets, virus infected computers, account for about 80% of spam.

E-mail addresses are collected from chatrooms, websites, newsgroups, and viruses which harvest users address books, and are sold to other spammers


Anti-Spam Techs

Some popular methods for filtering and refusing spam include e-mail filtering based on the content of the e-mail,

DNS-based blackhole lists (DNSBL), greylisting, spamtraps, enforcing technical requirements, checksumming systems to detect bulk email, and by putting some sort of cost on the sender via a Proof-of-work system or a micropayment.

Each method has strengths and weaknesses and each is controversial due to its weaknesses.

http://en.wikipedia.org/wiki/DNSBL

http://en.wikipedia.org/wiki/Greylisting

http://en.wikipedia.org/wiki/Spamtrap

http://en.wikipedia.org/wiki/Simple_Mail_Transfer_Protocol

http://en.wikipedia.org/wiki/Proof-of-work_system

http://en.wikipedia.org/wiki/Micropayment


Filtering Methods

Bayesian spam filtering CRM114 dSPAM Markovian discrimination POPFile Policyd-weight Postfix policy-daemon before SMTP DATA Procmail is an MDA (Mail Delivery Agent) for Unix systems. Maildrop is an MDA (Mail Delivery Agent) for Unix systems. Sendmail supports libmilter for mail filtering Sieve (mail filtering language) is an RFC standard for

describing mail filters SpamAssassin Anti-Spam SMTP Proxy information filtering White list#E-mail whitelists


Summary

have considered: secure email PGP S/MIME



Security on WWWXiang-Yang Li


Introduction

Introduction Presentation of SSL

• The inner workings of SSL• Attacks on SSL

Presentation of S-HTTP• Comparison with SSL/TLS• Attacks on S-HTTP

Other aspects of Web security• TLS• IPSec, Kerberos, SET

Conclusion


Web Security

Web now widely used by business, government, individuals

but Internet & Web are vulnerable have a variety of threats

integrity confidentiality denial of service authentication

need added security mechanisms


SSL (Secure Socket Layer)

transport layer security service originally developed by Netscape version 3 designed with public input subsequently became Internet standard

known as TLS (Transport Layer Security) uses TCP to provide a reliable end-to-

end service SSL has two layers of protocols


Location of SSL

Application Layer

Internet Protocol(IP)

Transmission Control Protocol (TCP)

Secure Socket Layer(SSL)

SSL is build on top of TCP

Provides a TCP like interface

In theory can be used by all type of applications in a transparent manner


SSL Architecture


SSL Architecture

SSL session an association between client & server created by the Handshake Protocol define a set of cryptographic parameters may be shared by multiple SSL connections

SSL connection a transient, peer-to-peer, communications link associated with 1 SSL session


SSL Record Protocol

confidentiality using symmetric encryption with a shared secret key

defined by Handshake Protocol IDEA, RC2-40, DES-40, DES, 3DES, Fortezza, RC4-

40, RC4-128 message is compressed before encryption

message integrity using a MAC with shared secret key similar to HMAC but with different padding


SSL Change Cipher Spec Protocol

one of 3 SSL specific protocols which use the SSL Record protocol

a single message causes pending state to become current hence updating the cipher suite in use


SSL Alert Protocol

conveys SSL-related alerts to peer entity severity

warning or fatal

specific alert unexpected message, bad record mac,

decompression failure, handshake failure, illegal parameter

close notify, no certificate, bad certificate, unsupported certificate, certificate revoked, certificate expired, certificate unknown

compressed & encrypted like all SSL data


SSL Handshake Protocol

allows server & client to: authenticate each other to negotiate encryption & MAC algorithms to negotiate cryptographic keys to be used

comprises a series of messages in phases Establish Security Capabilities Server Authentication and Key Exchange Client Authentication and Key Exchange Finish


General purpose

Two step process:• Handshake : exchange private keys using a public key encryption

algorithm• Data transmission: exchange the required data using a private key

encryption

`

1.Handshake

2. Data transmission


SSL Handshake Protocol


handshake

`

Client ServerClient Hello

Server Hello

Client Key Exchange

Change Cipher Specification

Server Certificate

Server Hello Done

Handshake Finished

Change Cipher Specifications

Handshake Finished


hello

Client “Hello”:• List of supported private

key encryptions +• Client random number

Server “Hello”: • Selected encryption

algorithm• Server Random number• Session ID

Server Certificate: • Verify server’s identity

`

Client Client Hello

Server Hello

Client Key Exchange


Server Certificate

Server Hello Done

Handshake Finished


Handshake Finished

Server


Key exchange

Client Key Exchange:• Client

Generate second random: Pre Master Key

Send Pre Master Key Calculate Master Key Calculate Secret Key Calculate MAC Key

• Server Calculate Master Key Calculate Secret Key Calculate MAC Key

`

Client Client Hello

Server Hello

Client Key Exchange


Server Certificate

Server Hello Done

Handshake Finished


Handshake Finished

Server


Resumed based on Session Id

`

Client ServerClient Hello

Server Hello


Handshake Finished


Handshake Finished


Certificate authority

Certificate Authority (CA) is a trusted third party that helps identify the server.

How does everything work?• Server sends ID, public key to CA• CA creates and signs the server’s Certificate• Client receives the Certificate from Server• Client verifies the Certificate using the signature and

the CA’s public key


MAC

MAC = Message Authentication Code The initial message is split into

fragments For each fragment a “fingerprint” is

calculated using the MAC key The fragment, fingerprint and record

header are encrypted and sent Receiver checks the “fingerprint” using

MAC key to detect inconsistent messages


Attacks on SSL

Certificate Injection Attack• The list of trusted Certificate Authorities is altered• Can be avoided by upgrading the OS or switching to a safer one.

Man in the Middle • Cipher Spec Rollback : regresses the public key encryption algorithms • Version Rollback : regression from SSL 3.0 to weaker SSL 2.0• Algorithm rollback : modify public encryption method• Truncation attack : TCP FIN|RST used to terminate connection

Timing attack• Can be avoided by randomly delaying the computations

Brute force• Can be used on servers that accept small key sizes: 40 bits for symmetric

encryptions and 512 for the asymmetric one.


TLS (Transport Layer Security)

IETF standard RFC 2246 similar to SSLv3 with minor differences

in record format version number uses HMAC for MAC a pseudo-random function expands secrets has additional alert codes some changes in supported ciphers changes in certificate negotiations changes in use of padding


TLS

TLS was developed by IETF to replace SSL version 3.

• Based on SSL version 3, with some changes:

• Replaced FORTEZZA key exchange option with DSS.

• Include the hash method HMAC used by IPSec for authentication in IP headers.

• More differentiation between sub-protocols.

• TLS has mechanisms for backwards compatibility with SSL.


TLS

TLS has about 30 possible cipher ‘suites’, combinations of key exchange, encryption method, and hashing method.

• Key exchange includes: RSA, DSS, Kerberos

• Encryption includes: IDEA(CBC), RC2, RC4, DES, 3DES, and AES

• Hashing: SHA and MD5

(Note: Some of the suites are intentionally weak export versions.)


Secure Electronic Transactions (SET) open encryption & security specification to protect Internet credit card

transactions developed in 1996 by Mastercard, Visa

etc not a payment system rather a set of security protocols &

formats secure communications amongst parties trust from use of X.509v3 certificates privacy by restricted info to those who need it


SET Components


SET Transaction

1. customer opens account2. customer receives a certificate3. merchants have their own certificates4. customer places an order5. merchant is verified6. order and payment are sent7. merchant requests payment authorization8. merchant confirms order9. merchant provides goods or service10. merchant requests payment


Dual Signature

customer creates dual messages order information (OI) for merchant payment information (PI) for bank

neither party needs details of other but must know they are linked use a dual signature for this

signed concatenated hashes of OI & PI


Purchase Request – Customer


Purchase Request – Merchant


Purchase Request – Merchant

1. verifies cardholder certificates using CA sigs2. verifies dual signature using customer's

public signature key to ensure order has not been tampered with in transit & that it was signed using cardholder's private signature key

3. processes order and forwards the payment information to the payment gateway for authorization (described later)

4. sends a purchase response to cardholder


Payment Gateway Authorization1. verifies all certificates2. decrypts digital envelope of authorization block to

obtain symmetric key & then decrypts authorization block

3. verifies merchant's signature on authorization block4. decrypts digital envelope of payment block to obtain

symmetric key & then decrypts payment block5. verifies dual signature on payment block6. verifies that transaction ID received from merchant

matches that in PI received (indirectly) from customer7. requests & receives an authorization from issuer8. sends authorization response back to merchant


Payment Capture

merchant sends payment gateway a payment capture request

gateway checks request then causes funds to be transferred to

merchants account notifies merchant using capture

response

Cryptography and Network Security 620Security on the WWW

C- Secure-HTTP

Presentation of S-HTTP

Designed by E. Rescorla and A. Schiffman of EIT to secure HTTP connections

Proposed in 1994 but never used commercially

Not to be confused with HTTPS: encrypts HTTP messages at the application level

ABCD


C- Secure-HTTP

Location of S-HTTPABCD

Internet Protocol(IP)

Transmission Control Protocol (TCP)

Secure-HTTPMessage encryption and

signature

Application Layer:HTTP message

HTTP-specific message encryption

Can possibly be used over a secure channel

Designed to be compatible with HTTP for handling at lower layers


C- Secure-HTTP

S-HTTP vs. SSL/TLS

HTTP-specific vs. general purpose SSL (IMAPS, POPS, LDAPS…)

Burden of encryption not on transmission/reception but rather on message production/unpacking

Similar set of available ciphers, plus added capabilities for signing (DSS, RSA)

Very general specifications, leaving a lot to implement and a potential for incompatible implementations

Only one reference implementation in NCSA Mosaic

ABCD


C- Secure-HTTP

S-HTTP vs. SSL/TLS: functionalitiesABCD

Security Service S-HTTP SSL

Privacy Public or private cryptosystemEncryption of the complete HTTP transaction

Symmetric key cryptosystemComplete communication encryption

Integrity Simple MAC or signing MAC only

Authentication Key management on the keys used, or digital signature

During the initial public key exchange (server auth. mandatory, client auth. optional)

Non-repudiation Digital signature Not provided

S-HTTP can make use of key management Non-repudiation is not provided by SSL Signing is optional, but a major attraction to S-HTTP


C- Secure-HTTP

S-HTTP vs. SSL/TLS: proxy traversalABCD

SSL-aware proxyExternal

secure server Enterprise environment

Proxy traversal: SSL connection

OR

S-HTTP-aware proxyExternal

secure server Enterprise environment

Proxy traversal: S-HTTP messaging

SSL tunnel SSL tunnel

Encrypted data

Authentication

cleartext


C- Secure-HTTP

S-HTTP inner working

Message-based encryption Superset of HTTP: “outer” envelope Specific headers added

ABCD

S-HTTP message

S-HTTP headers

HTTP payload headers:Security-Scheme, Encryption-Identity,Certificate-Info… + regular HTTP headers

HTTP message body

Request:Secure*Secure-HTTP/1.2

Response:Secure-HTTP/1.2 200 OK


C- Secure-HTTP

S-HTTP attacks

Basically the same as on SSL, since the ciphers are the same

Default values more secure in S-HTTP than SSL at the time of proposal (e.g. DES vs. RC4)

S-HTTP generally stronger by design (more resilient to proxy compromising)

More complex and wider specifications create a potential for faulty implementations

No real-world use to field test the actual security of S-HTTP

ABCD


D- Other protocols

HTTP has an authentication scheme as part of its original protocol.

HTTP Basic Authentication

ABCD

• Supported by almost all browsers and web servers.

• Password and username are sent in clear text (base64 encoded) in the HTTP request message.

• Obviously not secure enough for sensitive information.

This scheme is being replaced by the slightly more secure HTTP Digest Authentication, which sends a MD5 hash of the password and other information.


IPsec

IPSec is a security layer added to a computer’s protocol stack in the kernel (Below TCP). It is invisible to the application. It is implemented by adding additional protocol numbers in the IP protocol field.

• Good for implementing a VPN.

• Packets can be either tunneled inside IPSec packets, or Transported with only the data portion of the original packet encrypted.

• Every IPSec end machine (which could be a LAN’s router) must implement IPSec for it to work.


Summary

have considered: need for web security SSL/TLS transport layer security protocols

de facto standard, versatile and low-level enough to accommodate many types of payloads

SET secure credit card payment protocols IPSec: true network-layer security for any applications

(not just the Web) Kerberos: robust 2-way authentication framework with

emphasis on security manageability


Web Security

• SSL/TLS: de facto standard, versatile and low-level enough to accommodate many types of payloads

• S-HTTP: never took off, restricted to HTTP messages

• IPSec: true network-layer security for any applications (not just the Web)

• Kerberos: robust 2-way authentication framework with emphasis on security manageability

D- ConclusionABCD


Cryptography & Network Security

Malicious Software

XiangYang Li


Malicious Software

What is the concept of defense: The parrying of a blow. What is its characteristic feature: Awaiting the blow.—On War, Carl Von Clausewitz


Viruses and Other Malicious Content

computer viruses have got a lot of publicity

one of a family of malicious software effects usually obvious have figured in news reports, fiction,

movies (often exaggerated) getting more attention than deserve are a concern though


Malicious Software


Trapdoors

secret entry point into a program allows those who know access bypassing

usual security procedures have been commonly used by developers a threat when left in production programs

allowing exploited by attackers very hard to block in O/S requires good s/w development & update


Logic Bomb

one of oldest types of malicious software

code embedded in legitimate program activated when specified conditions met

eg presence/absence of some file particular date/time particular user

when triggered typically damage system modify/delete files/disks


Trojan Horse

program with hidden side-effects which is usually superficially attractive

eg game, s/w upgrade etc

when run performs some additional tasks allows attacker to indirectly gain access they do not

have directly

often used to propagate a virus/worm or install a backdoor

or simply to destroy data


Zombie

program which secretly takes over another networked computer

then uses it to indirectly launch attacks often used to launch distributed denial

of service (DDoS) attacks exploits known flaws in network

systems


Viruses

a piece of self-replicating code attached to some other code cf biological virus

both propagates itself & carries a payload carries code to make copies of itself as well as code to perform some covert task


Virus Operation

virus phases: dormant – waiting on trigger event propagation – replicating to programs/disks triggering – by event to execute payload execution – of payload

details usually machine/OS specific exploiting features/weaknesses


Virus Structure

program V :={goto main;1234567;subroutine infect-executable :={loop:

file := get-random-executable-file;if (first-line-of-file = 1234567) then goto loopelse prepend V to file; }

subroutine do-damage := {whatever damage is to be done}subroutine trigger-pulled := {return true if some condition holds}main: main-program := {infect-executable;

if trigger-pulled then do-damage;

goto next;}next:

}


Types of Viruses

can classify on basis of how they attack parasitic virus memory-resident virus boot sector virus stealth polymorphic virus macro virus


Macro Virus

macro code attached to some data file interpreted by program using file

eg Word/Excel macros esp. using auto command & command macros

code is now platform independent is a major source of new viral infections blurs distinction between data and

program files making task of detection much harder

classic trade-off: "ease of use" vs "security"


Email Virus

spread using email with attachment containing a macro virus cf Melissa

triggered when user opens attachment or worse even when mail viewed by

using scripting features in mail agent usually targeted at Microsoft Outlook

mail agent & Word/Excel documents


Worms

replicating but not infecting program typically spreads over a network

cf Morris Internet Worm in 1988 led to creation of CERTs

using users distributed privileges or by exploiting system vulnerabilities

widely used by hackers to create zombie PC's, subsequently used for further attacks, esp DoS

major issue is lack of security of permanently connected systems, esp PC's


Worm Operation

worm phases like those of viruses: dormant propagation

search for other systems to infect establish connection to target remote system replicate self onto remote system

triggering execution


Morris Worm

best known classic worm released by Robert Morris in 1988 targeted Unix systems using several propagation techniques

simple password cracking of local pw file exploit bug in finger daemon exploit debug trapdoor in sendmail daemon

if any attack succeeds then replicated self


Recent Worm Attacks

new spate of attacks from mid-2001 Code Red

exploited bug in MS IIS to penetrate & spread probes random IPs for systems running IIS had trigger time for denial-of-service attack 2nd wave infected 360000 servers in 14 hours

Code Red 2 had backdoor installed to allow remote control

Nimda used multiple infection mechanisms

email, shares, web client, IIS, Code Red 2 backdoor


Virus Countermeasures

viral attacks exploit lack of integrity control on systems

to defend need to add such controls typically by one or more of:

prevention - block virus infection mechanism detection - of viruses in infected system reaction - restoring system to clean state


Anti-Virus Software

first-generation scanner uses virus signature to identify virus or change in length of programs

second-generation uses heuristic rules to spot viral infection or uses program checksums to spot changes

third-generation memory-resident programs identify virus by actions

fourth-generation packages with a variety of antivirus techniques eg scanning & activity traps, access-controls


Advanced Anti-Virus Techniques

generic decryption use CPU simulator to check program signature &

behavior before actually running it

digital immune system (IBM) general purpose emulation & virus detection any virus entering org is captured, analyzed,

detection/shielding created for it, removed


Behavior-Blocking Software

integrated with host O/S monitors program behavior in real-time

eg file access, disk format, executable mods, system settings changes, network access

for possibly malicious actions if detected can block, terminate, or seek ok

has advantage over scanners but malicious code runs before

detection


Summary

have considered: various malicious programs trapdoor, logic bomb, trojan horse, zombie viruses worms countermeasures



Wireless LAN SecurityRoad to 802.11i

Xiangyang Li


Contents

Introduction Problem: 802.11b Not Secure! Wired Equivalent Privacy – WEP Types of Attacks 802.11b Proposed Solutions 802.1X Wi-Fi Protected Access (WPA) 802.11i: The Solution Conclusion


Introduction

Popular in offices, homes and public spaces (airport, coffee shop)

Most popular: 802.11b Example: Yahoo! DSL Wireless Kit Theoretical max @ 11Mbps Operate at 2.4GHz band DSSS/FSSS modulation – similar to CDMA phones


Introduction

Standards: IEEE 802.11 Series 802.11b – 11Mbps @ 2.4GHz 802.11a – 54Mbps @ 5.7GHz band 802.11g – 54Mbps @ 2.4GHz band 802.1X – security add-on 802.11i – high security


Problem: 802.11b Not Secure!

“ No inherent security” Wired Wireless media change was the objective

Wired Equivalent Privacy (WEP) The only “security” built into 802.11 Uses RC4 Stream Cipher – in a bad way Vulnerable to several types of attacks

Sometimes not even turned ON


Wired Equivalent Privacy – WEP

RC4 stream cipher Designed by Ron Rivest for RSA Security Very simple

Initialization Vector (IV) Shared Key

The issue is in the way RC4 is used IV (24 bits) reuse and fixed key Early versions used 40-bit key 128-bit mode effectively uses 104 bits


Wired Equivalent Privacy – WEP

RC4 Key Stream Encryption (source: http://mason.gmu.edu/~gharm/wireless.html)


Types of Attacks

AttacksConfidentialityIntegrityAvailability


Types of Attacks

Attacks on Confidentiality Traffic Analysis Passive Eavesdropping

Very easy to do Active Eavesdropping Unauthorized Access


Types of Attacks

Attacks on Confidentiality and/or Integrity Man-In-The-Middle

Attacks on Integrity Session Hijacking Replay

Attacks on Availability Denial of Service


802.11b Proposed Solutions

Virtual Private Network Closed Network

Through the use of SSID

Ethernet MAC address control lists Replace RC4 with block cipher Don’t reuse IV Automatic Key Assignment


802.1X: Interim Solution

Port-based authentication Not specific to wireless networks

Authentication servers RADIUS

Client authentication EAP


802.1X Problems

802.1X still has problems Extensible Authentication Protocol (EAP)

One-way authentication Attacks

Man-in-Middle Session Hijacking


802.1X Proposed Solutions

Per-packet authenticity and integrity Lots of overhead

Authenticity and integrity of EAPOL messages

Two-way authentication


Wi-Fi Protected Access (WPA)

Addresses issues with WEP Key management

TKIP Key expansion

Message Integrity Check

Software upgrade only Compatible with 802.1X Compatible with 802.11i


802.11i

Finalized: June, 2004 Robust Security Network Wi-Fi Alliance: WPA2 Improvements made

Authentication enhanced Key Management created Data Transfer security enhanced


802.11i - Authentication

Authentication Server Two-way authentication

Prevents man-in-the-middle attacks Master Key (MK) Pairwise Master Key (PMK)


802.11i – Key Management

Key Types Pairwise Transient Key Key Confirmation Key Key Encryption Key Group Transient Key Temporal Key


802.11i – Key Management

Source: http://csrc.nist.gov/wireless/S10_802.11i%20Overview-jw1.pdf

mailto:[email protected]


802.11i – Data Transfer

CCMP Long term solution – mandatory for 802.11i

compliance Latest AES encryption Requires hardware upgrades

WRAP Provided for early vendor support

TKIP Carried over from WPA


802.11i – Additional Enhancements Pre-authentication

Roaming clients

Client Validation Password-to-key mappings Random number generation


Conclusion

Basic 802.11b (with WEP) Massive security holes Easily attacked

802.1X Good interim solution Allows use of existing hardware Can still be attacked


Conclusion

Wi-Fi Protected Access Allows use of existing hardware Compatible with 802.1X Compatible with 802.11i

802.11i May require hardware upgrades Very secure Nothing is ever guaranteed



IPsec

XiangYang Li


IP Security

If a secret piece of news is divulged by a spy before the time is ripe, he must be put to death, together with the man to whom the secret was told.—The Art of War, Sun Tzu


IP Security

have considered some application specific security mechanisms eg. S/MIME, PGP, Kerberos, SSL/HTTPS

however there are security concerns that cut across protocol layers

would like security implemented by the network for all applications


IPSec

general IP Security mechanisms provides

authentication confidentiality key management

applicable to use over LANs, across public & private WANs, & for the Internet


IPSec Uses


Benefits of IPSec

in a firewall/router provides strong security to all traffic crossing the perimeter

is resistant to bypass is below transport layer, hence

transparent to applications can be transparent to end users can provide security for individual users

if desired


IP Security Architecture

specification is quite complex defined in numerous RFC’s

incl. RFC 2401/2402/2406/2408 many others, grouped by category

mandatory in IPv6, optional in IPv4


IPSec Services

Access control Connectionless integrity Data origin authentication Rejection of replayed packets

a form of partial sequence integrity

Confidentiality (encryption) Limited traffic flow confidentiality


Security Associations

a one-way relationship between sender & receiver that affords security for traffic flow

defined by 3 parameters: Security Parameters Index (SPI) IP Destination Address Security Protocol Identifier

has a number of other parameters seq no, AH & EH info, lifetime etc

have a database of Security Associations


Authentication Header (AH)

provides support for data integrity & authentication of IP packets end system/router can authenticate user/app prevents address spoofing attacks by tracking sequence

numbers

based on use of a MAC HMAC-MD5-96 or HMAC-SHA-1-96

parties must share a secret key


Authentication Header


Transport & Tunnel Modes


Encapsulating Security Payload (ESP)

provides message content confidentiality & limited traffic flow confidentiality

can optionally provide the same authentication services as AH

supports range of ciphers, modes, padding incl. DES, Triple-DES, RC5, IDEA, CAST etc CBC most common pad to meet blocksize, for traffic flow


Encapsulating Security Payload


Transport vs Tunnel Mode ESP

transport mode is used to encrypt & optionally authenticate IP data data protected but header left in clear can do traffic analysis but is efficient good for ESP host to host traffic

tunnel mode encrypts entire IP packet add new header for next hop good for VPNs, gateway to gateway security


Combining Security Associations

SA’s can implement either AH or ESP to implement both need to combine

SA’s form a security bundle

have 4 cases (see next)


Combining Security Associations


Key Management

handles key generation & distribution typically need 2 pairs of keys

2 per direction for AH & ESP

manual key management sysadmin manually configures every system

automated key management automated system for on demand creation of keys for

SA’s in large systems has Oakley & ISAKMP elements


Oakley

a key exchange protocol based on Diffie-Hellman key exchange adds features to address weaknesses

cookies, groups (global params), nonces, DH key exchange with authentication

can use arithmetic in prime fields or elliptic curve fields


ISAKMP

Internet Security Association and Key Management Protocol

provides framework for key management

defines procedures and packet formats to establish, negotiate, modify, & delete SAs

independent of key exchange protocol, encryption alg, & authentication method


ISAKMP


Summary

have considered: IPSec security framework AH ESP key management & Oakley/ISAKMP



Firewalls

XiangYang Li


Firewalls

The function of a strong position is to make the forces holding it practically unassailable—On War, Carl Von Clausewitz


Introduction

seen evolution of information systems now everyone want to be on the

Internet and to interconnect networks has persistent security concerns

can’t easily secure every system in org

need "harm minimisation" a Firewall usually part of this


What is a Firewall?

a choke point of control and monitoring interconnects networks with differing trust imposes restrictions on network services

only authorized traffic is allowed

auditing and controlling access can implement alarms for abnormal behavior

is itself immune to penetration provides perimeter defence


Firewall Limitations

cannot protect from attacks bypassing it eg sneaker net, utility modems, trusted organisations,

trusted services (eg SSL/SSH)

cannot protect against internal threats eg disgruntled employee

cannot protect against transfer of all virus infected programs or files because of huge range of O/S & file types


Firewalls – Packet Filters



simplest of components foundation of any firewall system examine each IP packet (no context)

and permit or deny according to rules hence restrict access to services (ports) possible default policies

that not expressly permitted is prohibited that not expressly prohibited is permitted




Attacks on Packet Filters

IP address spoofing fake source address to be trusted add filters on router to block

source routing attacks attacker sets a route other than default block source routed packets

tiny fragment attacks split header info over several tiny packets either discard or reassemble before check


Firewalls – Stateful Packet Filters

examine each IP packet in context keeps tracks of client-server sessions checks each packet validly belongs to one

better able to detect bogus packets out of context


Firewalls - Application Level Gateway (or Proxy)


Firewalls - Application Level Gateway (or Proxy)

use an application specific gateway / proxy

has full access to protocol user requests service from proxy proxy validates request as legal then actions request and returns result to user

need separate proxies for each service some services naturally support proxying others are more problematic custom services generally not supported


Firewalls - Circuit Level Gateway


Firewalls - Circuit Level Gateway

relays two TCP connections imposes security by limiting which such

connections are allowed once created usually relays traffic

without examining contents typically used when trust internal users

by allowing general outbound connections

SOCKS commonly used for this


Bastion Host

highly secure host system potentially exposed to "hostile" elements hence is secured to withstand this may support 2 or more net connections may be trusted to enforce trusted

separation between network connections runs circuit / application level gateways or provides externally accessible services


Firewall Configurations






Access Control

given system has identified a user determine what resources they can access general model is that of access matrix with

subject - active entity (user, process) object - passive entity (file or resource) access right – way object can be accessed

can decompose by columns as access control lists rows as capability tickets


Access Control Matrix


Trusted Computer Systems

information security is increasingly important have varying degrees of sensitivity of

information cf military info classifications: confidential, secret etc

subjects (people or programs) have varying rights of access to objects (information)

want to consider ways of increasing confidence in systems to enforce these rights

known as multilevel security subjects have maximum & current security level objects have a fixed security level classification


Bell LaPadula (BLP) Model

one of the most famous security models implemented as mandatory policies on system has two key policies: no read up (simple security property)

a subject can only read/write an object if the current security level of the subject dominates (>=) the classification of the object

no write down (*-property) a subject can only append/write to an object if the current security

level of the subject is dominated by (<=) the classification of the object


Reference Monitor


Evaluated Computer Systems

governments can evaluate IT systems against a range of standards:

TCSEC, IPSEC and now Common Criteria

define a number of “levels” of evaluation with increasingly stringent checking

have published lists of evaluated products though aimed at government/defense use can be useful in industry also


Summary

have considered: firewalls types of firewalls configurations access control trusted systems



Third Editionby William Stallings

Lecture slides by Lawrie Brown


Intruders

They agreed that Graham should set the test for Charles Mabledene. It was neither more nor less than that Dragon should get Stern's code. If he had the 'in' at Utting which he claimed to have this should be possible, only loyalty to Moscow Centre would prevent it. If he got the key to the code he would prove his loyalty to London Central beyond a doubt.—Talking to Strange Men, Ruth Rendell


Intruders

significant issue for networked systems is hostile or unwanted access

either via network or local can identify classes of intruders:

masquerader misfeasor clandestine user

varying levels of competence


Intruders

clearly a growing publicized problem from “Wily Hacker” in 1986/87 to clearly escalating CERT stats

may seem benign, but still cost resources

may use compromised system to launch other attacks


Intrusion Techniques

aim to increase privileges on system basic attack methodology

target acquisition and information gathering initial access privilege escalation covering tracks

key goal often is to acquire passwords so then exercise access rights of owner


Password Guessing

one of the most common attacks attacker knows a login (from email/web page etc) then attempts to guess password for it

try default passwords shipped with systems try all short passwords then try by searching dictionaries of common words intelligent searches try passwords associated with the user (variations on

names, birthday, phone, common words/interests) before exhaustively searching all possible passwords

check by login attempt or against stolen password file success depends on password chosen by user surveys show many users choose poorly


Password Capture

another attack involves password capture watching over shoulder as password is entered using a trojan horse program to collect monitoring an insecure network login (eg. telnet, FTP,

web, email) extracting recorded info after successful login (web

history/cache, last number dialed etc) using valid login/password can

impersonate user users need to be educated to use

suitable precautions/countermeasures


Intrusion Detection

inevitably will have security failures so need also to detect intrusions so can

block if detected quickly act as deterrent collect info to improve security

assume intruder will behave differently to a legitimate user but will have imperfect distinction between


Approaches to Intrusion Detection

statistical anomaly detection threshold profile based

rule-based detection anomaly penetration identification


Audit Records

fundamental tool for intrusion detection native audit records

part of all common multi-user O/S already present for use may not have info wanted in desired form

detection-specific audit records created specifically to collect wanted info at cost of additional overhead on system


Statistical Anomaly Detection

threshold detection count occurrences of specific event over time if exceed reasonable value assume intrusion alone is a crude & ineffective detector

profile based characterize past behavior of users detect significant deviations from this profile usually multi-parameter


Audit Record Analysis

foundation of statistical approaches analyze records to get metrics over time

counter, gauge, interval timer, resource use

use various tests on these to determine if current behavior is acceptable mean & standard deviation, multivariate, markov

process, time series, operational

key advantage is no prior knowledge used


Rule-Based Intrusion Detection

observe events on system & apply rules to decide if activity is suspicious or not

rule-based anomaly detection analyze historical audit records to identify usage

patterns & auto-generate rules for them then observe current behavior & match against rules to

see if conforms like statistical anomaly detection does not require prior

knowledge of security flaws


Rule-Based Intrusion Detection

rule-based penetration identification uses expert systems technology with rules identifying known penetration, weakness

patterns, or suspicious behavior rules usually machine & O/S specific rules are generated by experts who interview & codify

knowledge of security admins quality depends on how well this is done compare audit records or states against rules


Base-Rate Fallacy

practically an intrusion detection system needs to detect a substantial percentage of intrusions with few false alarms if too few intrusions detected -> false security if too many false alarms -> ignore / waste time

this is very hard to do existing systems seem not to have a

good record


Distributed Intrusion Detection

traditional focus is on single systems but typically have networked systems more effective defense has these

working together to detect intrusions issues

dealing with varying audit record formats integrity & confidentiality of networked data centralized or decentralized architecture


Distributed Intrusion Detection - Architecture


Distributed Intrusion Detection – Agent Implementation


Honeypots

decoy systems to lure attackers away from accessing critical systems to collect information of their activities to encourage attacker to stay on system so

administrator can respond

are filled with fabricated information instrumented to collect detailed

information on attackers activities may be single or multiple networked

systems


Password Management

front-line defense against intruders users supply both:

login – determines privileges of that user password – to identify them

passwords often stored encrypted Unix uses multiple DES (variant with salt) more recent systems use crypto hash function


Managing Passwords

need policies and good user education ensure every account has a default password ensure users change the default passwords to

something they can remember protect password file from general access set technical policies to enforce good

passwords minimum length (>6) require a mix of upper & lower case letters, numbers, punctuation block know dictionary words


Managing Passwords

may reactively run password guessing tools note that good dictionaries exist for almost any language/interest

group may enforce periodic changing of passwords have system monitor failed login attempts, &

lockout account if see too many in a short period

do need to educate users and get support balance requirements with user acceptance be aware of social engineering attacks


Proactive Password Checking

most promising approach to improving password security

allow users to select own password but have system verify it is acceptable

simple rule enforcement (see previous slide) compare against dictionary of bad passwords use algorithmic (markov model or bloom filter) to

detect poor choices


Summary

have considered: problem of intrusion intrusion detection (statistical & rule-based) password management


CryptographyNetSecurity-2008

Documents

network security 1cs549

othersinternet security

coursetextbookr cryptography

lecture note

noticethis lecture

yourselfrcan use library

office hoursofficersb

pages reportrfinal exam