Prime factorization

1
Prime factorization

• implementations in a functional language

2
Introduction
Introduction
Fermats algorithm
Pollards rho algorithm

• Goal
• Get a better understanding of the
  implementation and application of different
  factorization algorithms (Fermats Pollards
  rho Quadratic sieve Elliptic curve)

Elliptic curve factorization
Summary
3
Fermats algorithm
Introduction
Fermats algorithm

• Observation
• All composite numbers can be
  written as the difference between two squared
  numbers i.e.

Pollards rho algorithm
Elliptic curve factorization
Summary
4
Fermats algorithm
Introduction

• Algorithm
• Assume n is an odd number(otherwise factor out
  2 until is odd).
• Define
• Iteratively find .If is a
  square then and are factors
  of .If then stop and report
  as a prime.

Fermats algorithm
Pollards rho algorithm
Elliptic curve factorization
Summary
5
Fermats algorithm
Introduction
Fermats algorithm
Pollards rho algorithm
Elliptic curve factorization

• Is the algorithm correct
• Does it terminate

Summary
6
Fermats algorithm
Introduction

• Correctness
• The algorithm is correct iff
• Assume . Then

Fermats algorithm
Pollards rho algorithm
Elliptic curve factorization
Summary
Now assume . Then
Leading to the factor
7
Fermats algorithm
Introduction
Fermats algorithm
Pollards rho algorithm

• Termination
• Termination follows trivially from the
  fact that we iterate over a finite range.

Elliptic curve factorization
Summary
8
Fermats algorithm
Introduction

• Code

Fermats algorithm
(define (fermat-single n) (let ((s (get-sqrt
n)) (r (cdr s)) (m (- (expt r
2) n)) (r-stop (/ ( n 1) 2)))
(letrec ((iterator (lambda ()
(if (gt r r-stop)
(cons n ()) (begin
(set! s (get-sqrt m))
(if (car s) (cons
( r (cdr s)) (- r (cdr s)))
(begin (set! m
( m ( 2 r) 1))
(set! r ( r 1))
(iterator)))))))) (if (car s)
(cons r r) (iterator)))))
Pollards rho algorithm
Elliptic curve factorization
Summary
9
Fermats algorithm
Introduction

• Running times

Fermats algorithm
Pollards rho algorithm
Elliptic curve factorization
Summary
10
Pollards rho algorithm
Introduction
Fermats algorithm

• Observation
• If and are in different
  residue class modulo but in the same class
  modulo a proper divisor of then
  will result in a proper divisor of .

Pollards rho algorithm
Elliptic curve factorization
Summary
11
Pollards rho algorithm
Introduction

• Algorithm
• Choose a random function
• Define and
• Iteratively findIf then is a
  factorIf then go to step 1or report
  as maybe prime

Fermats algorithm
Pollards rho algorithm
Elliptic curve factorization
Summary
12
Pollards rho algorithm
Introduction
Fermats algorithm
Pollards rho algorithm
Elliptic curve factorization
Is the algorithm correct Does it terminate
Summary
13
Pollards rho algorithm
Introduction

• Correctness
• Since the range of is
  finitethe and values must cycle.It
  should be clear that cycles twice as fast as
  so if we go through a cycle with then
  so .If however
  then is a non-trivial factor of .

Fermats algorithm
Pollards rho algorithm
Elliptic curve factorization
Summary
14
Pollards rho algorithm
Introduction
Fermats algorithm

• Termination
• Termination follows from the
  cycling of the values and guaranteed termination
  when cycling has happened.

Pollards rho algorithm
Elliptic curve factorization
Summary
15
Pollards rho algorithm
Introduction

• Code

Fermats algorithm
(define (pollard-rho-single n) (let ((a 2)
(b 2) (c 1)) (letrec ((iterator
(lambda () (begin
(set! a (modulo ( (expt a 2) c)
n)) (set! b (modulo ( (expt b
2) c) n)) (set! b (modulo (
(expt b 2) c) n)) (let ((d (gcd
(- a b) n))) (cond ((and (gt d
1) (lt d n)) (cons d
(quotient n d))) (( d
n) (if ( c 2)
(cons n ())
(begin (set! a 2)
(set! b 2)
(set! c ( c 1))

(iterator)))) (else
(iterator)))))))) (iterator))))
Pollards rho algorithm
Elliptic curve factorization
Summary
16
Pollards rho algorithm
Introduction

• Running times

Fermats algorithm
The algorithm is too fasteven without
optimizationswhen the number has any small
factors (smaller than 10 digits). I have had
problems finding enough values to analyse
onthat give non-eligible running timesbut are
still feasible to factorize. (It factors
47189479742142798147947497147589257979528526917505
641 into3012764903 x 1566318025517134024710440446
4575395373798447in 25s)
Pollards rho algorithm
Elliptic curve factorization
Summary
17
Pollards rho algorithm
Introduction

• Running times

Fermats algorithm
Pollards rho algorithm
Elliptic curve factorization
Summary
18
Elliptic curve factorization
Introduction
Fermats algorithm

• Observation
• Iteratively applying a group function to a
  series of points starting on a random point in a
  group defined by an elliptic curve modulo the
  number we are factorizing we will eventually find
  a generator for the subgroup we iterate over.
  Using the order of this subgroup we can
  determine a factor of n.

Pollards rho algorithm
Elliptic curve factorization
Summary
19
Elliptic curve factorization
Introduction

• Code

Fermats algorithm
(define (elliptic-curve-single n) (let ((a 1)
(p (cons 0 5)) (e 2)) (letrec
((iterator (lambda ()
(begin (set! p (point-expt p e
a)) (set! e ( e 1))
(if (not (pair p)) (if
(symbol p) (cons n
()) (cons p (quotient n
p))) (iterator))))))
(iterator)))))
Pollards rho algorithm
Elliptic curve factorization
Summary
20
Elliptic curve factorization
Introduction

• Running times

Fermats algorithm
Pollards rho algorithm
Elliptic curve factorization
Summary
21
Summary
Introduction
Fermats algorithm
Pollards rho algorithm

• The following insight was gained through the
  project
• The elliptic curve algorithm is not fast in its
  natural form but becomes fast as elliptic
  curve knowledge is applied as optimizations.
• The implementation of the sieving process in
  quadratic sieve is complex and confusing
• A better understanding of the implemented
  algorithms

Elliptic curve factorization
Summary