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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
1
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.
2
Some values of (x)
3
The Sieve of Eratosthenes

This is an algorithm for generating all the primes up
to a given bound k.
4
The prime number theorem
5
The error term in the prime number
theory (1)
6
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
31
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
35
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)
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
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Example of D-H Key Exchange (2)


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Hybrid Encryption

Diffie-Hellman based hybrid encryption
system
A
K=(YB)xA
=(YA)xB
Mod q
SK=h(K)
128 – 256 bits
YA
B
YB
ESK(M)
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
riai=ria (mod qi)
=
h(p-1)/qi
mod p
a=kiqi+ai
ri= g(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
x  3 mod 4
x  0 mod 5
x  0 mod 7
x  8 mod 9
M / m1  1260 / 4  315
M / m2  1260 / 5  252
M / m3  1260 / 7  180
M / m4  1260 / 9  140
y1  3
y2  3
y3  3
y4  2
x  y1 ( M / m1 )c1  y2 ( M / m2 )c2  y3 ( M / m3 )c3  y4 ( M / m4 )c4
 3  315  3  3  252  0  3 180  0  2 140  8
 5075 mod 1260
 35
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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 Example 8.8
Malice select r U Fp*
Malice sends (c1, c2’=rc2) to Alice
Alice returns rm to Malice
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