implementations in a functional language Prime factorization implementations in a functional language

Introduction Introduction Fermat’s algorithm Pollard’s rho algorithm Goal: Get a better understanding of the implementation and application of different factorization algorithms (Fermat’s, Pollard’s rho, Quadratic sieve, Elliptic curve) Elliptic curve factorization Summary

Fermat’s algorithm Observation: Introduction Fermat’s algorithm Observation: All composite numbers can be written as the difference between two squared numbers, i.e. Pollard’s rho algorithm Elliptic curve factorization Summary

Fermat’s algorithm 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. Fermat’s algorithm Pollard’s rho algorithm Elliptic curve factorization Summary

Fermat’s algorithm Is the algorithm correct? Does it terminate? Introduction Fermat’s algorithm Pollard’s rho algorithm Elliptic curve factorization Is the algorithm correct? Does it terminate? Summary

Fermat’s algorithm Correctness: The algorithm is correct iff Introduction Correctness: The algorithm is correct iff Assume . Then Fermat’s algorithm Pollard’s rho algorithm Elliptic curve factorization Summary Now assume . Then Leading to the factor

Fermat’s algorithm Termination: Introduction Fermat’s algorithm Pollard’s rho algorithm Termination: Termination follows trivially from the fact that we iterate over a finite range. Elliptic curve factorization Summary

Fermat’s algorithm Code: Introduction Fermat’s algorithm Pollard’s rho (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 (>= r r-stop) (cons n '()) (begin (set! s (get-sqrt m)) (if (car s) (cons (+ r (cdr s)) (- r (cdr s))) (set! m (+ m (* 2 r) 1)) (set! r (+ r 1)) (iterator)))))))) (cons r r) (iterator))))) Pollard’s rho algorithm Elliptic curve factorization Summary

Fermat’s algorithm Running times: Introduction Fermat’s algorithm Pollard’s rho algorithm Elliptic curve factorization Summary

Pollard’s rho algorithm Introduction Fermat’s 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 . Pollard’s rho algorithm Elliptic curve factorization Summary

Pollard’s rho algorithm Introduction Algorithm: Choose a “random” function Define , , , and Iteratively find If then is a factor If then go to step 1 or report as “maybe prime” Fermat’s algorithm Pollard’s rho algorithm Elliptic curve factorization Summary

Pollard’s rho algorithm Introduction Fermat’s algorithm Pollard’s rho algorithm Elliptic curve factorization Is the algorithm correct? Does it terminate? Summary

Pollard’s 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 . Fermat’s algorithm Pollard’s rho algorithm Elliptic curve factorization Summary

Pollard’s rho algorithm Introduction Fermat’s algorithm Pollard’s rho algorithm Termination: Termination follows from the cycling of the values and guaranteed termination when cycling has happened. Elliptic curve factorization Summary

Pollard’s rho algorithm Introduction Code: Fermat’s 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)) (let ((d (gcd (- a b) n))) (cond ((and (> d 1) (< 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)))))))) Pollard’s rho algorithm Elliptic curve factorization Summary

Pollard’s rho algorithm Introduction Running times: Fermat’s algorithm ? The algorithm is too fast even without optimizations when the number has any “small” factors (smaller than 10 digits). I have had problems finding enough values to analyse on that give non-eligible running times, but are still feasible to factorize. (It factors 47189479742142798147947497147589257979528526917505641 into 3012764903 x 15663180255171340247104404464575395373798447 in 2,5s) Pollard’s rho algorithm Elliptic curve factorization Summary

Pollard’s rho algorithm Introduction Running times: Fermat’s algorithm Pollard’s rho algorithm Elliptic curve factorization Summary

Elliptic curve factorization Introduction Fermat’s 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. Pollard’s rho algorithm Elliptic curve factorization Summary

Elliptic curve factorization Introduction Code: Fermat’s 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))))) Pollard’s rho algorithm Elliptic curve factorization Summary

Elliptic curve factorization Introduction Running times: Fermat’s algorithm Pollard’s rho algorithm Elliptic curve factorization Summary

Summary The following insight was gained through the project Introduction Fermat’s algorithm Pollard’s rho algorithm The following insight was gained through the project The elliptic curve algorithm is not fast in it’s ”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