CSCE 715: Network Systems Security

CSCE 715:Network Systems Security

Chin-Tser [email protected]

University of South Carolina

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RSA Invented by Rivest, Shamir & Adleman of MIT in

1977 Best known and widely used public-key scheme Based on exponentiation in a finite (Galois)

field over integers modulo a prime exponentiation takes O((log n)3) operations

(easy) Use large integers (e.g. 1024 bits) Security due to cost of factoring large numbers

factorization takes O(e log n log log n) operations (hard)

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RSA Key Setup Each user generates a public/private key pair

by select two large primes at random: p, q compute their system modulus n=p·q

note ø(n)=(p-1)(q-1) select at random the encryption key e

where 1<e<ø(n), gcd(e,ø(n))=1 solve following equation to find decryption key d

e·d=1 mod ø(n) and 0≤d≤n publish their public encryption key: KU= {e,n} keep secret private decryption key: KR= {d,n}

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RSA Usage To encrypt a message M:

sender obtains public key of receiver KU={e,n}

computes: C=Me mod n, where 0≤M<n To decrypt the ciphertext C:

receiver uses its private key KR={d,n} computes: M=Cd mod n

Message M must be smaller than the modulus n (cut into blocks if needed)

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Why RSA Works Euler's Theorem:aø(n) mod n = 1 where gcd(a,n)=1

In RSA, we have n=p·q ø(n)=(p-1)(q-1) carefully chosen e and d to be inverses mod ø(n) hence e·d=1+k·ø(n) for some k

Hence :Cd = (Me)d = M1+k·ø(n) = M1·(Mø(n))k = M1·(1)k = M1 = M mod n

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RSA Example: Computing Keys

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 and e<160

choose e=7

5. Determine d: de=1 mod 160 and d<160 d=23 since 23×7=161=1×160+1

6. Publish public key KU={7,187}7. Keep secret private key KR={23,187}

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RSA Example: Encryption and Decryption

Given message M = 88 (88<187) Encryption:

C = 887 mod 187 = 11 Decryption:

M = 1123 mod 187 = 88

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Exponentiation Use a property of modular arithmetic[(a mod n)(b mod n)]mod n = (ab)mod n

Use the Square and Multiply Algorithm to multiply the ones that are needed to compute the result

Look at binary representation of exponent Only take O(log2 n) multiples for number n

e.g. 75 = 74·71 = 3·7 = 10 (mod 11) e.g. 3129 = 3128·31 = 5·3 = 4 (mod 11)

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RSA Key Generation Users of RSA must:

determine two primes at random - p,q select either e or d and compute the other

Primes p,q must not be easily derived from modulus n=p·q means p,q must be sufficiently large typically guess and use probabilistic test

Exponents e, d are multiplicative inverses, so use Inverse algorithm to compute the other

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Security of RSA

Four approaches to attacking RSA brute force key search (infeasible given

size of numbers) mathematical attacks (based on

difficulty of computing ø(n), by factoring modulus n)

timing attacks (on running of decryption) chosen ciphertext attacks (given

properties of RSA)

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Factoring Problem Mathematical approach takes 3 forms:

factor n=p·q, hence find ø(n) and then d determine ø(n) directly and find d find d directly

Currently believe all equivalent to factoring have seen slow improvements over the years

as of May-05 best is 200 decimal digits (663 bits) with LS biggest improvement comes from improved algorithm

cf “Quadratic Sieve” to “Generalized Number Field Sieve” to “Lattice Sieve”

1024+ bit RSA is secure barring dramatic breakthrough ensure p, q of similar size and matching other constraints

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Timing Attacks Developed in mid-1990’s Exploit timing variations in operations

e.g. multiplying by small vs large number Infer operand size based on time taken RSA exploits time taken in

exponentiation Countermeasures

use constant exponentiation time add random delays blind values used in calculations

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Chosen Ciphertext Attacks RSA is vulnerable to a Chosen Ciphertext

Attack (CCA) attackers chooses ciphertexts and gets

decrypted plaintext back choose ciphertext to exploit properties of

RSA to provide info to help cryptanalysis can counter with random pad of plaintext or use Optimal Asymmetric Encryption

Padding (OAEP)

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Key Management

Asymmetric encryption helps address key distribution problems

Two aspects distribution of public keys use of public-key encryption to

distribute secret keys

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Distribution of Public Keys

Four alternatives of public key distribution Public announcement Publicly available directory Public-key authority Public-key certificates

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Public Announcement Users distribute public keys to

recipients or broadcast to community at large E.g. 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’s and broadcast it

can masquerade as claimed user before forgery is discovered

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Publicly Available Directory Achieve 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

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Public-Key Authority Improve security by tightening control

over distribution of keys from directory Has properties of directory Require users to know public key for the

directory Users can interact with directory to

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

keys are needed

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Public-Key Authority

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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, authorized rights, etc With all contents signed by a trusted

Public-Key or Certificate Authority (CA) Can be verified by anyone who knows

the CA’s public key

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Public-Key Certificates

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Distribute Secret KeysUsing Asymmetric Encryption

Can use previous methods to obtain public key of other party

Although public key can be used for confidentiality or authentication, asymmetric encryption algorithms are too slow

So usually want to use symmetric encryption to protect message contents

Can use asymmetric encryption to set up a session key

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Simple Secret Key Distribution Proposed by Merkle in 1979

A generates a new temporary public key pair A sends B the public key and A’s identity B generates a session key Ks and sends

encrypted Ks (using A’s public key) to A A decrypts message to recover Ks and both use

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Problem with Simple Secret Key Distribution

An adversary can intercept and impersonate both parties of protocol

A generates a new temporary public key pair {KUa, KRa} and sends KUa || IDa to B

Adversary E intercepts this message and sends KUe || IDa to B

B generates a session key Ks and sends encrypted Ks (using E’s public key)

E intercepts message, recovers Ks and sends encrypted Ks (using A’s public key) to A

A decrypts message to recover Ks and both A and B unaware of existence of E

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Next Class

Key exchange Diffie-Hellman key exchange

protocol Elliptic curve cryptography Read Chapters 11 and 12