1. implementations in a functional language
Prime factorization
implementations in a functional language

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

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

10. 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

11. 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

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

13. 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

14. 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

15. 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

16. 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
into x in 2,5s)
Pollard’s rho
algorithm
Elliptic curve
factorization
Summary

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

18. 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

19. 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

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

21. 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