Number Theory and Advanced Cryptography 2

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Presentation transcript:

Number Theory and Advanced Cryptography 2 Number Theory and Advanced Cryptography 2. Primes and Discrete Logarithms Part I: Introduction to Number Theory Part II: Advanced Cryptography Chih-Hung Wang Feb. 2011

The distribution of primes The natural way of measuring the density of primes is to count the number of primes up to a bound x, where x is a real number. For a real number x ¸ 0, the function (x) is defined to be the number of primes up to x. Thus, (1) = 0, (2) = 1, (7.5) = 4, and so on.

Some values of (x)

The Sieve of Eratosthenes This is an algorithm for generating all the primes up to a given bound k.

The prime number theorem

The error term in the prime number theory (1)

The error term in the prime number theory (2)

Sophie Germain primes

Probabilistic primality testing Trial Division

Trial division

The Miller-Rabin test

Error parameter (1)

Error parameter (2)

Carmichael numbers

Good Primality testing (1)

Good Primality testing (2)

Error parameter

Generating random primes using the Miller-Rabin Test

Sieving up to a small bound

Generating a random k-bit prime

Perfect power testing (1)

Perfect power testing (2)

Perfect power testing (3)

Deterministic Primality Testing The basic idea

AKS algorithm

Running time

Notes

Primality testing in Java Public BigInteger ( int bitLength，int certainty，Random rnd ) Public boolean isProbablePrime (int certainty)

Cyclic groups Order of group element

Order of group element

(Example)Powers of Integers, Modulo 19

Cyclic group & Group generator

Example of Cyclic Group

Theorem of Cyclic Group

Prime Order group

The Multiplicative Group Zn*

The Multiplicative Group Zn*

Example of The Multiplicative Group

Finding Primitive Root Page 166

Application 1: Diffie-Hellman Key Exchange Diffie and Hellman 1976 A number of commercial products employ this key exchange technique This algorithm enables two users to exchange key securely

The Diffie-Hellman Key Exchange Protocol

Example of D-H Key Exchange (1) q=97 =5 XA = 36 XB=58 YA=536=50 mod 97 YB=558=44 mod 97 K=(YB)XA mod 97 = 4436 = 75 nod 97 K=(YA)XB mod 97 = 5058 = 75 nod 97

Example of D-H Key Exchange (2)

Hybrid Encryption Diffie-Hellman based hybrid encryption system A B YA K=(YB)xA =(YA)xB Mod q SK=h(K) YB ESK(M) 128 – 256 bits SK can be a key of the AES symmetric cryptosystem

The Man-in-the-Middle Attack (1)

The Man-in-the-Middle Attack (2)

The DH Problem and DL Problem (1)

The DH Problem and DL Problem (2) Example: a = loggh = log3 5 mod 19 = 4

Importance of Arbitrary Instances for Intractability Assumptions CRT a=kiqi+ai ri= g(p-1)/qi mod p riai=ria (mod qi) = h(p-1)/qi mod p

Chinese Remainder Theorem (1)

Chinese Remainder Theorem (2)

Chinese Remainder Theorem (3)

Example of CRT

ElGamal (1)

ElGamal (2)

Meet-in-the-middle attack & Active attack of ElGamal See Page 277 Example 8.8 Malice select Malice sends (c1, c2’=rc2) to Alice Alice returns rm to Malice