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;;;; "factor.scm", prime test and factorization for Scheme
;;; Copyright (C) 1991, 1992, 1993 Aubrey Jaffer.
;
;Permission to copy this software, to redistribute it, and to use it
;for any purpose is granted, subject to the following restrictions and
;understandings.
;
;1. Any copy made of this software must include this copyright notice
;in full.
;
;2. I have made no warrantee or representation that the operation of
;this software will be error-free, and I am under no obligation to
;provide any services, by way of maintenance, update, or otherwise.
;
;3. In conjunction with products arising from the use of this
;material, there shall be no use of my name in any advertising,
;promotional, or sales literature without prior written consent in
;each case.
(require 'random)
(require 'modular)
;;; (modulo p 16) is because we care only about the low order bits.
;;; The odd? tests are inline of (expt -1 ...)
(define (prime:jacobi-symbol p q)
(cond ((zero? p) 0)
((= 1 p) 1)
((odd? p)
(if (odd? (quotient (* (- (modulo p 16) 1) (- q 1)) 4))
(- (prime:jacobi-symbol (modulo q p) p))
(prime:jacobi-symbol (modulo q p) p)))
(else
(let ((qq (modulo q 16)))
(if (odd? (quotient (- (* qq qq) 1) 8))
(- (prime:jacobi-symbol (quotient p 2) q))
(prime:jacobi-symbol (quotient p 2) q))))))
;;;; Solovay-Strassen Prime Test
;;; if n is prime, then J(a,n) is congruent mod n to a**((n-1)/2)
;;; See:
;;; Robert Solovay and Volker Strassen,
;;; "A Fast Monte-Carlo Test for Primality,"
;;; SIAM Journal on Computing, 1977, pp 84-85.
;;; checks if n is prime. Returns #f if not prime. #t if (probably) prime.
;;; probability of a mistake = (expt 2 (- prime:trials))
;;; choosing prime:trials=30 should be enough
(define prime:trials 30)
;;; prime:product is a product of small primes.
(define prime:product
(let ((p 210))
(for-each (lambda (s) (set! p (or (string->number s) p)))
'("2310" "30030" "510510" "9699690" "223092870"
"6469693230" "200560490130"))
p))
(define (prime:prime? n)
(set! n (abs n))
(cond ((<= n 36) (and (memv n '(2 3 5 7 11 13 17 19 23 29 31)) #t))
((= 1 (gcd n prime:product))
(do ((i prime:trials (- i 1))
(a (+ 1 (random (- n 1))) (+ 1 (random (- n 1)))))
((not (and (positive? i)
(= (gcd a n) 1)
(= (modulo (prime:jacobi-symbol a n) n)
(modular:expt n a (quotient (- n 1) 2)))))
(if (positive? i) #f #t))))
(else #f)))
;;;;Lankinen's recursive factoring algorithm:
;From: ld231782@longs.LANCE.ColoState.EDU (L. Detweiler)
; | undefined if n<0,
; | (u,v) if n=0,
;Let f(u,v,b,n) := | [otherwise]
; | f(u+b,v,2b,(n-v)/2) or f(u,v+b,2b,(n-u)/2) if n odd
; | f(u,v,2b,n/2) or f(u+b,v+b,2b,(n-u-v-b)/2) if n even
;Thm: f(1,1,2,(m-1)/2) = (p,q) iff pq=m for odd m.
;It may be illuminating to consider the relation of the Lankinen function in
;a `computational hierarchy' of other factoring functions.* Assumptions are
;made herein on the basis of conventional digital (binary) computers. Also,
;complexity orders are given for the worst case scenarios (when the number to
;be factored is prime). However, all algorithms would probably perform to
;the same constant multiple of the given orders for complete composite
;factorizations.
;Thm: Eratosthenes' Sieve is very roughtly O(ln(n)/n) in time and
; O(n*log2(n)) in space.
;Pf: It works with all prime factors less than n (about ln(n)/n by the prime
; number thm), requiring an array of size proportional to n with log2(n)
; space for each entry.
;Thm: `Odd factors' is O((sqrt(n)/2)*log2(n)) in time and O(log2(n)) in
; space.
;Pf: It tests all odd factors less than the square root of n (about
; sqrt(n)/2), with log2(n) time for each division. It requires only
; log2(n) space for the number and divisors.
;Thm: Lankinen's algorithm is O(sqrt(n)/2) in time and O((sqrt(n)/2)*log2(n))
; in space.
;Pf: The algorithm is easily modified to seach only for factors p<q for all
; pq=m. Then the recursive call tree forms a geometric progression
; starting at one, and doubling until reaching sqrt(n)/2, or a length of
; log2(sqrt(n)/2). From the formula for a geometric progression, there is
; a total of about 2^log2(sqrt(n)/2) = sqrt(n)/2 calls. Assuming that
; addition, subtraction, comparison, and multiplication/division by two
; occur in constant time, this implies O(sqrt(n)/2) time and a
; O((sqrt(n)/2)*log2(n)) requirement of stack space.
(define (prime:f u v b n)
(if (<= n 0)
(cond ((negative? n) #f)
((= u 1) #f)
((= v 1) #f)
; Do both of these factors need to be factored?
(else (append (or (prime:f 1 1 2 (quotient (- u 1) 2))
(list u))
(or (prime:f 1 1 2 (quotient (- v 1) 2))
(list v)))))
(if (even? n)
(or (prime:f u v (+ b b) (quotient n 2))
(prime:f (+ u b) (+ v b) (+ b b) (quotient (- n (+ u v b)) 2)))
(or (prime:f (+ u b) v (+ b b) (quotient (- n v) 2))
(prime:f u (+ v b) (+ b b) (quotient (- n u) 2))))))
(define (prime:factor m)
(if
(negative? m) (cons -1 (prime:factor (- m)))
(let* ((s (gcd m prime:product))
(r (quotient m s)))
(if (even? s)
(append
(if (= 1 r) '() (prime:factor r))
(cons 2 (let ((s/2 (quotient s 2)))
(if (= s/2 1) '()
(or (prime:f 1 1 2 (quotient (- s/2 1) 2))
(list s/2))))))
(if (= 1 s) (or (prime:f 1 1 2 (quotient (- m 1) 2)) (list m))
(append (if (= 1 r) '()
(or (prime:f 1 1 2 (quotient (- r 1) 2)) (list r)))
(or (prime:f 1 1 2 (quotient (- s 1) 2)) (list s))))))))
(define jacobi-symbol prime:jacobi-symbol)
(define prime? prime:prime?)
(define factor prime:factor)
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