; quadratic sieve
; f: maximum prime in factor base
; 2*m: number of entries in sieve
; relations with x in car and ys in cdr
; examples
; (qs 87463 30 30) => 587
; (qs 13290059 150 300) => 4261
; (qs (* 2971215073 99194853094755497) 2000 3000000) => 99194853094755497
(define verbose? #t)
(define (primes n)
(let ((bits (make-vector (+ n 1) #t)))
(let loop ((p 2) (ps '()))
(cond ((< n p) (reverse ps))
((vector-ref bits p)
(do ((i (+ p p) (+ i p))) ((< n i))
(vector-set! bits i #f))
(loop (+ p 1) (cons p ps)))
(else (loop (+ p 1) ps))))))
(define prime?
(let ((seed 3141592654))
(lambda (n)
(define (rand)
(set! seed (modulo (+ (* 69069 seed) 1234567) 4294967296))
(+ (quotient (* seed (- n 2)) 4294967296) 2))
(define (expm b e m)
(define (times x y) (modulo (* x y) m))
(let loop ((b b) (e e) (r 1))
(if (zero? e) r
(loop (times b b) (quotient e 2)
(if (odd? e) (times b r) r)))))
(define (spsp? n a)
(do ((d (- n 1) (/ d 2)) (s 0 (+ s 1)))
((odd? d)
(let ((t (expm a d n)))
(if (or (= t 1) (= t (- n 1))) #t
(do ((s (- s 1) (- s 1))
(t (expm t 2 n) (expm t 2 n)))
((or (zero? s) (= t (- n 1)))
(positive? s))))))))
(if (not (integer? n))
(error 'prime? "must be integer")
(if (< n 2) #f
(do ((a (rand) (rand)) (k 25 (- k 1)))
((or (zero? k) (not (spsp? n a)))
(zero? k))))))))
(define (square x) (* x x))
(define (isqrt n)
(if (not (and (positive? n) (integer? n)))
(error 'isqrt "must be positive integer")
(let loop ((x n))
(let ((y (quotient (+ x (quotient n x)) 2)))
(if (< y x) (loop y) x)))))
(define (expm b e m)
(define (m* x y) (modulo (* x y) m))
(cond ((zero? e) 1)
((even? e) (expm (m* b b) (/ e 2) m))
(else (m* b (expm (m* b b) (/ (- e 1) 2) m)))))
(define (mod-sqrt a p)
(define (both n) (list n (- p n)))
(cond ((not (and (odd? p) (prime? p)))
(error 'mod-sqrt "modulus must be an odd prime"))
((not (= (jacobi a p) 1))
(error 'mod-sqrt "must be a quadratic residual"))
(else (let ((a (modulo a p)))
(case (modulo p 8)
((3 7) (both (expm a (/ (+ p 1) 4) p)))
((5) (let* ((x (expm a (/ (+ p 3) 8) p))
(c (expm x 2 p)))
(if (= a c) (both x)
(both (modulo (* x (expm 2 (/ (- p 1) 4) p)) p)))))
(else (let* ((d (let loop ((d 2))
(if (= (jacobi d p) -1) d
(loop (+ d 1)))))
(s (let loop ((p (- p 1)) (s 0))
(if (odd? p) s
(loop (quotient p 2) (+ s 1)))))
(t (quotient (- p 1) (expt 2 s)))
(big-a (expm a t p))
(big-d (expm d t p))
(m (let loop ((i 0) (m 0))
(cond ((= i s) m)
((= (- p 1)
(expm (* big-a (expm big-d m p))
(expt 2 (- s 1 i)) p))
(loop (+ i 1) (+ m (expt 2 i))))
(else (loop (+ i 1) m))))))
(both (modulo (* (expm a (/ (+ t 1) 2) p)
(expm big-d (/ m 2) p)) p)))))))))
(define (msqrt a p) ; principal (smaller) value of modular square root
(apply min (mod-sqrt a p)))
(define (jacobi a m)
(if (not (integer? a)) (error 'jacobi "must be integer")
(if (not (and (integer? m) (positive? m) (odd? m)))
(error 'jacobi "modulus must be odd positive integer")
(let loop1 ((a (modulo a m)) (m m) (t 1))
(if (zero? a) (if (= m 1) t 0)
(let ((z (if (member (modulo m 8) (list 3 5)) -1 1)))
(let loop2 ((a a) (t t))
(if (even? a) (loop2 (/ a 2) (* t z))
(loop1 (modulo m a) a
(if (and (= (modulo a 4) 3)
(= (modulo m 4) 3))
(- t) t))))))))))
(define (factor-base n f)
(let loop ((ps (cdr (primes f))) (fs (list 2)))
(cond ((null? ps) (reverse fs))
((= (jacobi n (car ps)) 1)
(loop (cdr ps) (cons (car ps) fs)))
(else (loop (cdr ps) fs)))))
(define (smooth n fb) ; list of factors with -1, or null if not smooth
(let ((sign (if (negative? n) -1 1)) (n (abs n)))
(let loop ((n n) (fb fb) (fs (list)))
(cond ((null? fb) (list))
((= n 1) (if (negative? sign) (cons -1 (reverse fs)) (reverse fs)))
((zero? (modulo n (car fb)))
(loop (/ n (car fb)) fb (cons (car fb) fs)))
(else (loop n (cdr fb) fs))))))
(define (qs n f m)
(let* ((e 10) ; fudge factor on sum of logarithms
(sqrt-n (isqrt n)) (b (- sqrt-n m)) (fb (factor-base n f))
(sieve (make-vector (+ m m) (- e (inexact->exact (round (log (* 2 sqrt-n)))))))
(ts (map (lambda (f) (msqrt n f)) (cdr fb))) ; exclude 2
(ls (map (lambda (f) (inexact->exact (round (log f)))) (cdr fb))))
(when verbose? (display "Factor base of ") (display (length fb))
(display " primes") (newline))
(do ((fb (cdr fb) (cdr fb)) (ts ts (cdr ts)) (ls ls (cdr ls))) ((null? fb))
(do ((i (modulo (- (car ts) b) (car fb)) (+ i (car fb)))) ((<= (+ m m) i))
(vector-set! sieve i (+ (vector-ref sieve i) (car ls))))
(do ((i (modulo (- (- (car ts)) b) (car fb)) (+ i (car fb)))) ((<= (+ m m) i))
(vector-set! sieve i (+ (vector-ref sieve i) (car ls)))))
(let loop ((i 0) (rels (list)))
(if (= i (+ m m))
(begin
(when verbose? (display "Found ") (display (length rels))
(display " smooth relations") (newline))
(solve n fb rels))
(if (positive? (vector-ref sieve i))
(let ((ys (smooth (- (square (+ i b)) n) fb)))
(if (pair? ys)
(loop (+ i 1) (cons (cons (+ i b) ys) rels))
(loop (+ i 1) rels)))
(loop (+ i 1) rels))))))
(define (solve n fb rels)
(display rels) (newline))
(define (make-expo-vector fb rel)
(define (add-1bit x) (if (zero? x) 1 0))
(let loop ((fb fb) (rel rel) (prev -2) (es (list)))
(cond ((null? fb) (list->vector (reverse es)))
((null? rel) (loop (cdr fb) rel prev (cons 0 es)))
((= (car rel) prev) (loop fb (cdr rel) prev (cons (add-1bit (car es)) (cdr es))))
((= (car rel) (car fb)) (loop (cdr fb) (cdr rel) (car fb) (cons 1 es)))
(else (loop (cdr fb) rel (car fb) (cons 0 es))))))
(define (make-identity-matrix n)
(let ((id (make-vector n)))
(do ((i 0 (+ i 1))) ((= i n) id)
(let ((v (make-vector n 0)))
(vector-set! v i 1)
(vector-set! id i v)))))
(define (right-most-one vec r)
(let ((row (vector-ref vec r)))
(let loop ((i (- (vector-length row) 1)))
(if (negative? i) i
(if (= (vector-ref row i) 1) i
(loop (- i 1)))))))
(define (pivot-row expo c)
(let ((max-r (vector-length expo)))
(let loop ((r 0))
(if (= r max-r) r
(if (= (right-most-one expo r) c) r
(loop (+ r 1)))))))
(define (add-rows matrix r1 r2)
(define (add a b) (if (= a b) 0 1))
(let ((row1 (vector-ref matrix r1)) (row2 (vector-ref matrix r2)))
(do ((i 0 (+ i 1))) ((= i (vector-length row1)) row2)
(vector-set! row2 i (add (vector-ref row1 i) (vector-ref row2 i))))))
(define (any-one? vec r)
(let* ((row (vector-ref vec r)) (r-len (vector-length row)))
(let loop ((i 0))
(if (= i r-len) #f
(if (positive? (vector-ref row i)) #t
(loop (+ i 1)))))))
(define (factor n hist rels r)
(let* ((h (vector-ref hist r))
(h-len (vector-length h)))
(let loop ((i 0) (x 1) (y2 1))
(if (= i h-len)
(let ((g (gcd (- x (isqrt y2)) n)))
(if (< 1 g n) g #f))
(if (= (vector-ref h i) 1)
(loop (+ i 1)
(* x (car (vector-ref rels i)))
(apply * y2 (cdr (vector-ref rels i))))
(loop (+ i 1) x y2))))))
(define (solve n fb rels)
(let* ((fb (cons -1 fb)) (fb-len (length fb)) (rel-len (length rels))
(expo (list->vector (map (lambda (rel) (make-expo-vector fb (cdr rel))) rels)))
(hist (make-identity-matrix rel-len))
(rels (list->vector rels)))
(do ((c (- fb-len 1) (- c 1))) ((negative? c))
(let ((p (pivot-row expo c)))
(do ((r (+ p 1) (+ r 1))) ((<= rel-len r))
(when (= (right-most-one expo r) c)
(vector-set! expo r (add-rows expo p r))
(vector-set! hist r (add-rows hist p r))))))
(let loop ((r 0))
(cond ((= r rel-len) #f)
((any-one? expo r) (loop (+ r 1)))
((factor n hist rels r) =>
(lambda (f) (if f f (loop (+ r 1)))))
(else (loop (+ r 1)))))))
(display (qs 13290059 150 300))
programmingpraxis
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on Mar 9: