COEN 350: Network Security Overview of Cryptography
Dec 25, 2015
COEN 350: Network Security
Overview of Cryptography
Overview of Cryptography
Table of contents Introduction Cryptographic Security One Way Functions Secret Key Cryptography Public Key Cryptography Message Authentication Codes Zero Knowledge Proofs Diffie Hellman Key Exchange
Cryptography Traditional use of cryptography
Encrypt a plain text into cypher Only people with the right knowledge can
recover plain text. Secret Key (Symmetric) Cryptography
Encryption and decryption use secret key c. Public Key (Asymmetric) Cryptography
Encryption and decryption use two different keys.
Cryptography
Other uses of cryptography Secure data while stored. Authenticate entities. Ensure integrity of data. Sign statements so that signature
cannot be repudiated.
Cryptography
Other uses of cryptography Fast file destruction:
Encrypt files with a secret key. Destroy secret key to securely delete the
file. E-cash
Hash Functions Given an object, create a hash
(short bit-string) of the object. Hashs differ Objects differ Objects differ with overwhelming prob.
Hashes differ Cryptographically secure hash:
Given a hash, cannot find object with that hash.
Hash Functions
Tripwire Protect OS against trojans. Maintain hashes of all system libraries
in a secure area. Check hash against known hash
periodically.
Overview of Cryptography
Table of contents Introduction Cryptographic Security One Way Functions Secret Key Cryptography Public Key Cryptography Message Authentication Codes Zero Knowledge Proofs Diffie Hellman Key Exchange
Cryptographic Security
Leverage in cryptography comes from functions that are hard to compute without special knowledge.
“Hard to compute” difficult to substantiate
Cryptographic Security “Hard to compute” = NP complete
Problem is P: can be solved deterministically in polynomial time.
Problem is NP: solution can be verified in polynomial time.
Central Conjecture: NP P. NP-complete: If this problem can be solved in
polynomial time then all NP problems can be solved in polynomial time.
NP-complete problems: Intrinsically difficult problems to solve on a computer.
But: NP completeness is tendency. Instances of NP-complete problems can be easy
to solve. Knapsack problem.
Cryptographic Security
“Computationally hard” = “Takes n years to solve on best machine.”
Breaking codes is usually parallelizable. Use distributed attack. SETI@home
Moore’s law: Computers double in speed every 16 months.
Cryptographic Security
UNIX password cracking UNIX passwords are 8 characters long.
Assume 102 printable characters in a password.
1016 possible passwords. 10000 password attempts a second takes
1012/2 seconds to find random password. 16,000 years to find password
Dictionary attacks take much less.
Cryptographic Security DES Data encryption standard Published in 1977 by National Bureau of
Standards. Uses 56 bit key Brute-Force attack succeeds after ~1016 tries. 1977: Diffie Hellman:
Spend $20,000,000.- to build parallel machine that can find key in 12 hours.
1998: Electronic Frontier Association Build DES cracker for $250,000.- that could break a
key in 4 days. $150,000.- for second cracker
Cryptographic Security
Security of Algorithms Fundamental Security Paradigm
"If a lot of smart people have tried to crack a paradigm for a long time, then it is impossible to crack the
paradigm."
Cryptographic SecurityModels for evaluating security Unconditional Security
Adversary has unlimited computational resources, but there is not enough information available to defeat the system.
Example: One Time Pad Complexity Theoretic Security
Defines an appropriate model of computation Adversaries can mount attacks that use space and
time polynomial resources. These attacks might be in practice impossible. True attacks might be non-polynomial.
Cryptographic Security
Models for evaluating security Provable Security
Difficulty of defeating a protocol is at least as hard as another (supposedly difficult) problem.
Computational Security Measures the amount of effort (using
the best methods available now) required to defeat a system.
Overview of Cryptography
Table of contents Introduction Cryptographic Security One Way Functions Secret Key Cryptography Public Key Cryptography Message Authentication Codes Zero Knowledge Proofs Diffie Hellman Key Exchange
One-Way Functions
One way function Easy to compute Hard to invert.
“Hard” means computationally infeasible.
One-Way Functions Example X = {1, 2, ... , 16} Define f: X → X, x → x3 mod 17.
This function is reasonably easy to compute.
Surprisingly hard to calculate logarithms in a finite field.
Use the following table. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
1 8 10 13 6 12 3 2 15 14 5 11 4 7 9 16l
One-Way Functions
Pre-image resistance: Given a possible image y, it is
computationally impossible to find any preimage x such that f (x) = y.
Second pre-image resistance: Given a pre-image x, it is
computationally infeasible to find another preimage z, z x, such that f (x) = f (z).
One-Way Functions
Collision resistant: It is computationally infeasible to find
any two distincts inputs x, x', x' x such that f(x) = f(x').
One-Way Functions
Definition: A function f is a strong one-way hash function (also known as a collision resistant (one-way) hash function) if and only if f is easily computable, that is, given x, it is
easy to calculate f(x).
f is pre-image resistant.
f is second pre-image resistant.
f is collision resistant.
One-Way Functions
One-Way function with trapdoors Much in cryptography is based on
being able to do a difficult thing when possessing a secret.
There are one-way functions that are easy to invert if one knows a secret.
One-Way Functions Choose
p = 48611 (a prime) q = 53993 (a prime) n = p·q.
Define f f (x) = x 3 mod n. f is one way, if we only know n. If we know the secret that n = pq, then
there is an algorithm that solves x 3 = y mod n for given y and unknown x.
One-Way Functions One-way function with trapdoor
Family of functions fi where i I, an index set.
Each fi is one-way. There exists functions hi and a secret
s such that hi (s, .) is easy to compute fi (hi (s, y)) = y.
That is, hi (s, .) is the inverse function of fi
Overview of Cryptography
Table of contents Introduction Cryptographic Security One Way Functions Secret Key Cryptography Public Key Cryptography Message Authentication Codes Zero Knowledge Proofs Diffie Hellman Key Exchange
Secret Key Cryptography
Conventional encryption uses a secret to convert plaintext to cipher and the same secret to convert cipher to plaintext.
A Greek general tattoos the message into the crown of the head of a slave who then lets his hair grow again. When the slave reaches the destination, the recipient reads the message after the slave has shaven his head again.
One-time pad Caesar’s cypher
Secret Key Cryptography
Encryption uses an algorithm publicly known.
Sender and receiver use a secret key.
Secret Key Cryptography
Generic recipe: Take the plain text. Apply a transformation (based on secret,
reversible with secret). Repeat until result is sufficiently
disguised Product cipher
Use first one transformation, then another one.
Secret Key Cryptography
Substitution Permutation Network Each state involves substitutions
and permutations. Substitutions:
Take an input, replace it by an output. Often implemented as a table.
Input needs to be small.
Secret Key Cryptography Permutations
Take the bits and reorder them.
Secret Key Cryptography Substitution
Permutation Network
Encode from top to bottom
Decode from bottom to top
Secret Key Cryptography Iterated block cipher
Made up of rounds. In each round, apply an transformation
with a separate key (the round key). Feistel Cipher
Secret Key Cryptography Feistel Cipher
Iterated Block cipher Block size is 2t. Each round:
Breaks input into left half L(n) and right half R(n)
L(n+1) = R(n). R(n+1) = Mangler(R(n), Kn) L(n)
Kn is round key.
Secret Key Cryptography
Feistel round for encryption (left) and decryption (right)
Secret Key Cryptography DES (1977)
uses a 64b key with a parity check, so that effective key size is 56b.
Derives 16 round keys of 48b each. Works on input of size 64. Uses 16 round Feistel algorithm
IDEA (1991) Uses a 128b key Uses 8 computationally identical rounds based on
generalized Feistel algorithm Additional beginning and ending transformation.
Secret Key Cryptography Typical block code takes 64b plaintext
and changes it to 64b cipher text. Electronic Code Book:
Break plain text into 64b-blocks. Encrypt all blocks. Vulnerable to attacks
Two identical text blocks are encrypted the same way.
Allows guessing contents. Reordering of plain text = Reordering of cipher
text. Change meaning of cipher text.
Secret Key Cryptography Example:
Database contains employee and salary information.
Encrypted:
Secret Key Cryptography Switch portion of cipher text
Resulting plaintext
Secret Key CryptographyCipher Block Chaining
Encryption and Decryption
Secret Key Cryptography Cipher Block Chaining If we do not mind to mangle some
data, we can switch bits. How? Your turn.
Secret Key Cryptography
Assume we want to flip bit 3 in m4 We switch bit 3 in c3
This probably mangles m3 But has the desired effect on m4
Secret Key Cryptography Second thread to CBC:
Assume attacker knows plain text and cipher, i.e. m1, m2, …, c1, c2, …, IV
Attacker can calculate D(c1), D(c2), … Can build library of ci D(ci) and use it for
other attacks.
Secret Key Cryptography Output Feedback modes Same idea, but prevents these types
of attacks.
Output Feed Back Cipher Feed Back
Secret Key Cryptography One-Time Pad
Only proven secure cryptographic method But the pad needs to be transmitted
between sender and receiver. XORing with a short string is not
secure. See projects
Secret Key Cryptography
RC4 One time pad generated by random
number generator, seeded with key Considered still secure (if you let the
RNG run for a few hundred rounds) If plain-text can be guessed,
vulnerable to bit flipping How? (Your turn)
Secret Key Cryptography Message Authentication Code
Can be calculated with cipher block chaining or similar method.
c6 is the MAC
Overview of Cryptography
Table of contents Introduction Cryptographic Security One Way Functions Secret Key Cryptography Public Key Cryptography Message Authentication Codes Zero Knowledge Proofs Diffie Hellman Key Exchange
Public Key Cryptography Asymmetric Key Cryptograpy.
Use one key for encryption, another for decryption.
E(e,.) encryption with key e D(d,.) is decryption with key d D(d,E(e,m)) = E(e,D(d,m)) = m for all
messages m. Note: Not all public key systems have this
commutativity between D and E.
Public Key Cryptography Keep one key public, the other one private. Use public key to encrypt, give Bob secret key to
decrypt.
Public Key Cryptography Signing Messages.
Alice creates a public key pair (e,d) and gives or publishes e.
Alice uses private key to calculate s = D(d,m) Pad with zeroes if necessary.
Bob uses Alice's public key to decrypt E(e,s) = E(e,D(d,m')) = m.
If m is in the format that a signed message has, then Bob accepts the message as truly Alice's.
Public Key CryptographyRSA RSA: Rivest Shamir Adleman Choose n = pq, p, q large primes Select e that is coprime to –
(n)=(p – 1)(q – 1) Find d such that ed = 1 mod (n).
Only computationally feasible if n = pq is known.
Public key: (e,n) Private key: (d,n)
Public Key CryptographyRSA Encryption with private key. Divide messages into chunks < n Encrypt chunk c as c1 = ce mod n. Decryption
Calculate c = c1d mod n.
c1d = (ce)d = ced = cx(n)+1 = c
Using Euler’s theorem a(n) = 1 mod n.
Public Key Cryptography
RSA is safe if used with caution.
Overview of Cryptography
Table of contents Introduction Cryptographic Security One Way Functions Secret Key Cryptography Public Key Cryptography Message Authentication Codes Zero Knowledge Proofs Diffie Hellman Key Exchange
Message Authentication Code
Also known as MIC (Message Integrity Code).
Append MAC to message. Nobody can change message
without changing MAC. Easy to check whether MAC
belongs to the message.
Message Authentication Code
Symmetric key MACs using a hash function Message Calculate hash value Protect hash value by encrypting it
with a secret key. Sender and receiver share the
secret key.
Message Authentication Code ISO 8732-2: (Banking - Approved Algorithm for Message
Authentication)
MAC(Message M) { for(i=0; i <= LengthOfMessage; i++) {
v = v << 1 e = v XOR w x = [ ((e + y mod 2**32) or A and C) * (x XOR M(i) ] mod 2**32-1 y = [ ((e + x mod 2**32) or B and D) * (y XOR M(i) ] mod 2**32-2 } return x XOR y;
}
where A, B, C, and D are constants, and v and w are determined by the key.
Message Authentication Code Cipher Block Chaining (CBC) derived
MAC
Message Authentication Code Public Key Message Authentication
If message is small Alice encrypts with private key. Bob decrypts with public key.
If message looks right, it comes from someone who knows Alice’s key.
If message is large Calculate a digest (hash) of the message Alice encrypts digest with private key. Bob decrypts digest with public key. If digest matches, it comes from Alice.
Message Authentication Code MD5
MD5 was developed by Rivest in 1994. Its 128 bit (16 byte) message digest
SHA1 Secure Hash Algorithm developed by NIST published in 1994 produces a 160-bit (20 byte) message
digest
Message Authentication Codes
MD5 is broken: Possible to generate two meaningful
files with the same MD5 value. SHA-1 is almost broken:
Finding a file with a given SHA-1 value is now computationally feasible.
Use SHA-256, SHA-512, WHIRLPOOL
Overview of Cryptography
Table of contents Introduction Cryptographic Security One Way Functions Secret Key Cryptography Public Key Cryptography Message Authentication Codes Zero Knowledge Proofs Diffie Hellman Key Exchange
Zero Knowledge Proof A potentially new way for identification. Challenge-Response
Claimant proves identity to verifier by demonstrating knowledge of a secret.
E.g. Verifier gives Claimant random number. Claimant can encode it, thus proving
knowledge of a secret key. But observer now knows an encryption with
the secret key: a knowledge leak Password: Forces claimant to give out
password.
Zero-Knowledge Proofs
Interactive proof system that claimant knows secret.
But secret is not revealed to verifier or an observer.
Zero-Knowledge Proof
Alice wants to convince Bob that she knows the secret word to the door in this cave.
But Alice doesn’t want to show Bob how she does it.
Zero-Knowledge Proof Bob and Alice
walk to A. Alice walks to
either C or D. Bob goes to B and
cries out “left” or “right”
Alice can satisfy the request.
Zero-Knowledge Proof Repeat n times. Alice is lucky with
probability ½ and does not have to open door.
Alice is always lucky with probability 1/2n
Zero-Knowledge Proof Hostile observer
cannot distinguish between Alice and Bob playing a charade or Alice knowing how to get through the door.
Zero Knowledge Proof. Fiat Shamir:
A trusted center publishes a modulus n = p·q that is the product of two primes.
Alice selects a secret s coprime to n and publishes v = s 2 mod n as its public key.
Repeat n times: Alice chooses a random number r and sends r2 to
Bob. Bob randomly selects e = 0 or e = 1. Alice sends y = r · se mod n to Bob Bob accepts this proof if y 2 = r 2 v e mod n.
Zero Knowledge Proof
Assume Alice is an impostor. If Alice guesses that Bob will send
e = 0. Alice picks s and sends v = s2 to Bob. Bob asks for e = 0. Alice sends rs Bob checks out that (rs)2 = r 2 v
Zero Knowledge Proof
Assume Alice is an impostor. If Alice guesses that Bob will send
e = 1. Alice picks a and sends v = a2/v to
Bob. Bob asks for e = 1. Alice sends a Bob checks out that a 2 = a 2/v · v
Overview of Cryptography
Table of contents Introduction Cryptographic Security One Way Functions Secret Key Cryptography Public Key Cryptography Message Authentication Codes Zero Knowledge Proofs Diffie Hellman Key Exchange
Diffie Hellmann
First public key system. Two partners share secret number. No eavesdropper can deduce
secret number.
Diffie Hellman p large prime g < p
Best choice is a generator modulo p: n i : n = gi.
(p,g) are public. Alice picks secret r. Bob picks secret s. Alice sends gr mod p to Bob. Bob sends gs mod p to Alice. Common key is t = (gr)s = (gs)r mod t.
Diffie Hellman
Snooper needs to derive t from gr and gs.
This is computationally equivalent to calculate r from gr mod p.
Diffie Hellman Man-in-the-middle-attack:
Mallory intercepts communications between Alice and Bob.
Alice sends gr to Mallory. Mallory sends gr’ to Bob. Bob sends gs to Mallory. Mallory sends gs’ to Alice. Alice establishes common key grs’ with Mallory. Bob establishes common key gr’s with Mallory. Alice and Bob communicate via Mallory, who
reads the traffic (or changes it).
Diffie Hellman
Social defense against this attack: Alice publicly distributes gr mod p.