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Prime factorization

- implementations in a functional language

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

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

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

Fermats algorithm

Introduction

Fermats algorithm

Pollards rho algorithm

Elliptic curve factorization

- Is the algorithm correct?
- Does it terminate?

Summary

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

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

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

Fermats algorithm

Introduction

- Running times

Fermats algorithm

Pollards rho algorithm

Elliptic curve factorization

Summary

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

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

Pollards rho algorithm

Introduction

Fermats algorithm

Pollards rho algorithm

Elliptic curve factorization

Is the algorithm correct? Does it terminate?

Summary

Pollards rho algorithm

Introduction

- Correctness
- Since the range of is

finite,the 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

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

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

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 times,but are

still feasible to factorize. (It factors

47189479742142798147947497147589257979528526917505

641 into3012764903 x 1566318025517134024710440446

4575395373798447in 2,5s)

Pollards rho algorithm

Elliptic curve factorization

Summary

Pollards rho algorithm

Introduction

- Running times

Fermats algorithm

Pollards rho algorithm

Elliptic curve factorization

Summary

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

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

Elliptic curve factorization

Introduction

- Running times

Fermats algorithm

Pollards rho algorithm

Elliptic curve factorization

Summary

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

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