Outline
Definitions
Properties of primes
Millard-Rabin Primality Test
Pollard’s Rho Heuristic
Primality test and Factorization
The Millard-Rabin primality test and the Rho-Pollard
factorization heuristic
Alpar Sadr
Carleton University
March 31, 2015
Alpar Sadr
Primality test and Factorization
Outline
Definitions
Properties of primes
Millard-Rabin Primality Test
Pollard’s Rho Heuristic
Definitions
Basics
Primality
Properties of primes
Fermat’s Theorem
Pseudoprimes
Distribution of numbers
Millard-Rabin Primality Test
Millard-Rabin Algorithm
The Composite Function
Pollard’s Rho Heuristic
Integer Factorization
Algorithm
References
Alpar Sadr
Primality test and Factorization
Outline
Definitions
Properties of primes
Millard-Rabin Primality Test
Pollard’s Rho Heuristic
Basics
Primality
Definitions
Let a, b and c be integers ( a, b, c ∈ Z ).
I
a divides b if there is an integer k such that b = ak.
I
a is a divisor of b if a > 0.
The trivial divisors of a are 1 and a.
I
a is a common divisor of b and c
if a is a divisor of b and a is a divisor of c.
I
The greatest common divisor of a and b, denoted by
GCD(a, b), is the maximum of the set of common divisors of
a and b.
Alpar Sadr
Primality test and Factorization
Outline
Definitions
Properties of primes
Millard-Rabin Primality Test
Pollard’s Rho Heuristic
Basics
Primality
Definitions
Let a, b ∈ Z.
I
a is a prime if its only divisors are the trivial ones.
I
a is a composite if it has other divisors.
I
a and b are coprime if GCD(a, b) = 1.
Note that there are numbers that are neither prime nor composite,
such as 0 or 1.
Alpar Sadr
Primality test and Factorization
Outline
Definitions
Properties of primes
Millard-Rabin Primality Test
Pollard’s Rho Heuristic
Fermat’s Theorem
Pseudoprimes
Distribution of numbers
Fermat’s Little Theorem
Theorem
If p is prime, then for any integer a, p divides ap − a
Alpar Sadr
Primality test and Factorization
Outline
Definitions
Properties of primes
Millard-Rabin Primality Test
Pollard’s Rho Heuristic
Fermat’s Theorem
Pseudoprimes
Distribution of numbers
Implications of Fermat’s Little Theorem
Theorem: p divides ap − a for every integer a
I
Theorem can be rewritten as ap ≡ a mod p.
I
If a and p are coprime then ap−1 ≡ 1 mod p.
I
If for some integer a, p does not divide ap − a, then p is not
prime.
Alpar Sadr
Primality test and Factorization
Outline
Definitions
Properties of primes
Millard-Rabin Primality Test
Pollard’s Rho Heuristic
Fermat’s Theorem
Pseudoprimes
Distribution of numbers
Implications of Fermat’s Little Theorem
Theorem: p divides ap − a for every integer a
I
Theorem can be rewritten as ap ≡ a mod p.
I
If a and p are coprime then ap−1 ≡ 1 mod p.
I
If for some integer a, p does not divide ap − a, then p is not
prime.
I
Can we use this fact to find prime numbers?
Alpar Sadr
Primality test and Factorization
Outline
Definitions
Properties of primes
Millard-Rabin Primality Test
Pollard’s Rho Heuristic
Fermat’s Theorem
Pseudoprimes
Distribution of numbers
Pseudoprimes and Charmichael numbers
Strong Theorem: If a and p are coprime then ap−1 ≡ 1 mod p.
I
There are composite numbers that satisfy ap−1 ≡ 1 mod p for
some integer a.
Those are base-a pseudoprimes.
Alpar Sadr
Primality test and Factorization
Outline
Definitions
Properties of primes
Millard-Rabin Primality Test
Pollard’s Rho Heuristic
Fermat’s Theorem
Pseudoprimes
Distribution of numbers
Pseudoprimes and Charmichael numbers
Strong Theorem: If a and p are coprime then ap−1 ≡ 1 mod p.
I
There are composite numbers that satisfy ap−1 ≡ 1 mod p for
some integer a.
Those are base-a pseudoprimes.
I
What about trying different bases?
Alpar Sadr
Primality test and Factorization
Outline
Definitions
Properties of primes
Millard-Rabin Primality Test
Pollard’s Rho Heuristic
Fermat’s Theorem
Pseudoprimes
Distribution of numbers
Pseudoprimes and Charmichael numbers
Strong Theorem: If a and p are coprime then ap−1 ≡ 1 mod p.
I
There are composite numbers that satisfy ap−1 ≡ 1 mod p for
some integer a.
Those are base-a pseudoprimes.
I
What about trying different bases?
I
There are composite numbers that satisfy ap−1 ≡ 1 mod p for
every integer a.
Those are the Charmichael numbers or Fermat pseudoprimes.
Alpar Sadr
Primality test and Factorization
Outline
Definitions
Properties of primes
Millard-Rabin Primality Test
Pollard’s Rho Heuristic
Fermat’s Theorem
Pseudoprimes
Distribution of numbers
Distribution of primes
Theorem
The prime number theorem:
lim
π(n)
n→∞
n
ln(n)
=1
where π(n) is the number of primes up to n.
I
This gives a fair approximation of the distribution of prime
numbers over an interval.
I
The difference in the ratio between numerator and
denominator is of 6% for n = 109 .
Alpar Sadr
Primality test and Factorization
Outline
Definitions
Properties of primes
Millard-Rabin Primality Test
Pollard’s Rho Heuristic
Fermat’s Theorem
Pseudoprimes
Distribution of numbers
Distribution of Charmichael numbers
I
Lower bound on the number of Charmichael numbers up to
√
some large integer n is of 3 n.
Note that 1021 is considered small as the lower bound is not
satisfied yet.
I
The known upper bound is ugly but grows very slowly.
We know how many times we should sample numbers to get a
possible prime and since Charmichael numbers are rare, trying
different bases should increase our confidence.
Alpar Sadr
Primality test and Factorization
Outline
Definitions
Properties of primes
Millard-Rabin Primality Test
Pollard’s Rho Heuristic
Millard-Rabin Algorithm
The Composite Function
Algorithm
Given an odd number n > 2 and a positive integer s,
MILLER-RABIN(n, s)
for j ← 1 to s
a ← random(1, n − 1)
if ISCOMPOSITE(a, n) then return COMPOSITE
return PRIME
The error rate of the Miller-Rabin Algorithm for any s > 0 and odd
integer n > 2 is at most of 2−s .
Alpar Sadr
Primality test and Factorization
Outline
Definitions
Properties of primes
Millard-Rabin Primality Test
Pollard’s Rho Heuristic
Millard-Rabin Algorithm
The Composite Function
The ISCOMPOSITE method
The ISCOMPOSITE(a, n) has two main components.
1. Checks if n is composite using Fermat’s Little Theorem
2. Checks if n is composite using the following Lemma
Lemma
If there exists an integer a such that a2 ≡ 1 mod n
then n is composite.
Alpar Sadr
Primality test and Factorization
Outline
Definitions
Properties of primes
Millard-Rabin Primality Test
Pollard’s Rho Heuristic
Millard-Rabin Algorithm
The Composite Function
Modular Exponentiation
Given two integers a and b, there is a simple way to find the result
of ai mod b for any i.
Example
Let a = 3 and b = 11,
i
0
1
2
3
4
5
6
7
3i
1
3
9
27
81
243
729
2187
3i mod 11
1
3
9
5
4
1
3
9
Alpar Sadr
Primality test and Factorization
Outline
Definitions
Properties of primes
Millard-Rabin Primality Test
Pollard’s Rho Heuristic
Integer Factorization
Algorithm
References
Quick Introduction
I
Factoring an integer n into prime factors can be done with
√
trial division by all integers up to n.
Alternatively, factoring an integer n2 would be achieved with
trial division by all integers up to n.
I
Pollard devised a heuristic that is expected to factor any
number up to n4 for the same amount of work.
Alpar Sadr
Primality test and Factorization
Outline
Definitions
Properties of primes
Millard-Rabin Primality Test
Pollard’s Rho Heuristic
Integer Factorization
Algorithm
References
Pollard’s Rho Heureistic’s Algorithm
POLLARD-RHO(n)
i←1
x ← random(0, n − 1)
y[i] ← x
k←2
while TRUE
x ← (x2 − 1) mod n
d ← GCD(y[i] − x, n)
i←i+1
if d 6= 1 and d 6= n
print d
if i = k
y[i] ← x
k ← 2k
Alpar Sadr
for j ← 1 to s
Primality test and Factorization
Outline
Definitions
Properties of primes
Millard-Rabin Primality Test
Pollard’s Rho Heuristic
Integer Factorization
Algorithm
References
References
I
Thomas H. Cormen et al.. Introduction to Algorithms, Second
edition. Chapters 31.1,31.6,31.8 and 31.9.
I
Wikipedia for Charmichael numbers and Fermat’s Little
Theorem.
I
PCTeX’s introductory TeX sample to Beamer found at
http://pctex.com/UsingBeamer.html
Alpar Sadr
Primality test and Factorization