Number Theory and Advanced Cryptography 2' Primes

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Number Theory and Advanced Cryptography 2.
Primes Discrete Logarithms --Primes
distribution testing, DL background
applications
Part I Introduction to Number Theory Part II
Advanced Cryptography

• Chih-Hung Wang
• Feb. 2006

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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, ?(75) 4, and so on.

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Some values of ?(x)
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The Sieve of Eratosthenes

• This is an algorithm for generating all the
  primes up to a given bound k.

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The prime number theorem
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The error term in the prime number theory (1)
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The error term in the prime number theory (2)
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Sophie Germain primes
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Probabilistic primality testing

• Trial Division

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Trial division
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The Miller-Rabin test
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Error parameter (1)
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Error parameter (2)
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Carmichael numbers
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Good Primality testing (1)
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Good Primality testing (2)
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Error parameter
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Generating random primes using the Miller-Rabin
Test
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Sieving up to a small bound
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Generating a random k-bit prime
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Perfect power testing (1)
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Perfect power testing (2)
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Perfect power testing (3)
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Deterministic Primality Testing

• The basic idea

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AKS algorithm
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Running time
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Notes
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Primality testing in Java

• Public BigInteger ( int bitLength,int
  certainty,Random rnd )
• Public boolean isProbablePrime (int certainty)

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Cyclic groups

• Order of group element

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Order of group element
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(Example)Powers of Integers, Modulo 19
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Cyclic group Group generator
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Example of Cyclic Group
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Theorem of Cyclic Group
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Prime Order group
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The Multiplicative Group Zn
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The Multiplicative Group Zn
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Example of The Multiplicative Group
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Finding Primitive Root
Page 166
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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

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The Diffie-Hellman Key Exchange Protocol
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Example of D-H Key Exchange (1)
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XA 36 XB58
q97
YA53650 mod 97 YB55844 mod 97
K(YB)XA mod 97 4436 75 nod 97 K(YA)XB mod
97 5058 75 nod 97
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Example of D-H Key Exchange (2)
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Hybrid Encryption

• Diffie-Hellman based hybrid encryption system

A
B
YA
K(YB)xA (YA)xB Mod q SKh(K)
YB
ESK(M)
128 256 bits
SK can be a key of the AES symmetric cryptosystem
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The Man-in-the-Middle Attack (1)
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The Man-in-the-Middle Attack (2)
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The DH Problem and DL Problem (1)
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The DH Problem and DL Problem (2)
Example a loggh log3 5 mod 19 4
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Importance of Arbitrary Instances for
Intractability Assumptions
CRT
akiqiai ri g(p-1)/qi mod p
riairia (mod qi) h(p-1)/qi mod p
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Chinese Remainder Theorem (1)
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Chinese Remainder Theorem (2)
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Chinese Remainder Theorem (3)
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Example of CRT
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ElGamal (1)
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ElGamal (2)
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Meet-in-the-middle attack Active attack of
ElGamal

• See Page 277