Multiple Polynomial Quadratic Sieve
June 21, 2013
; multiple polynomial quadratic sieve
(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 (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 (ilog b n)
(let loop1 ((lo 0) (b^lo 1) (hi 1) (b^hi b))
(if (< b^hi n) (loop1 hi b^hi (* hi 2) (* b^hi b^hi))
(let loop2 ((lo lo) (b^lo b^lo) (hi hi) (b^hi b^hi))
(if (<= (- hi lo) 1) (if (= b^hi n) hi lo)
(let* ((mid (quotient (+ lo hi) 2))
(b^mid (* b^lo (expt b (- mid lo)))))
(cond ((< n b^mid) (loop2 lo b^lo mid b^mid))
((< b^mid n) (loop2 mid b^mid hi b^hi))
(else mid))))))))
(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 (lift-root n p) ; hensel's lemma
(let* ((r (apply min (mod-sqrt n p)))
(s (/ (- (modulo n (* p p)) (* r r)) p))
(t (modulo (* (inverse (+ r r) p) s) p))
(u (+ r (* t p))))
(list u (- (* p p) u))))
(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 (inverse x m)
(let loop ((x x) (a 0) (b m) (u 1))
(if (positive? x)
(let ((q (quotient b x)) (r (remainder b x)))
(loop (modulo b x) u x (- a (* q u))))
(if (= b 1) (modulo a m) (error 'inverse "must be coprime")))))
(define (factor-base n f) ; => (values fb ts ls e fb-len)
(let loop ((ps (cdr (primes f))) (fb (list 2)) (ts (list)) (ls (list)) (x 2) (limit 5))
(when (and (pair? ps) (< limit (car ps)))
(set! x (+ x 1)) (set! limit (isqrt (expt 2 (+ x x 1)))))
(cond ((null? ps) (values (reverse fb) (reverse ts) (reverse ls) (max 5 x) (length fb)))
((= (jacobi n (car ps)) 1)
(loop (cdr ps) (cons (car ps) fb)
(cons (apply min (mod-sqrt n (car ps))) ts) (cons x ls) x limit))
(else (loop (cdr ps) fb ts ls x limit)))))
(define (smooth n fb) ; list of smooth factors of n in descending order, or null
(let loop ((n (abs n)) (fb fb) (fs (if (negative? n) (list -1) (list))))
(cond ((null? fb) (list))
((< n (* (car fb) (car fb))) (cons n fs))
((zero? (modulo n (car fb)))
(loop (/ n (car fb)) fb (cons (car fb) fs)))
(else (loop n (cdr fb) fs)))))
(define (sieve n fb ts ls e f m a b q-inv) ; => (values rels parts)
(define (make-rel x ys)
(cons (modulo (* (+ (* a x) b) q-inv) n) ys))
; a relation, whether full or partial, has (ax+b)/q in its car and a
; list of factors of g(x)=ax^2+2bx+c, in descending order, in its cdr;
; a large prime, if it exists, is at the cadr of the relation
(let* ((c (/ (- (* b b) n) a)) (rels (list)) (parts (list))
(sieve (make-vector (+ m m 1) (+ e e))))
(do ((fb (cdr fb) (cdr fb)) (ts ts (cdr ts)) (ls ls (cdr ls))
(invs (map (lambda (f) (inverse a f)) (cdr fb)) (cdr invs)))
((null? fb))
(let ((f (car fb)) (t (car ts)) (l (car ls)) (v (car invs)))
(do ((i (modulo (* v (- t b)) f) (+ i f))) ((<= (+ m m 1) i))
(vector-set! sieve i (+ (vector-ref sieve i) l)))
(do ((i (modulo (* v (- (- t) b)) f) (+ i f))) ((<= (+ m m 1) i))
(vector-set! sieve i (+ (vector-ref sieve i) l)))))
(do ((i 0 (+ i 1))) ((= (+ m m 1) i))
(let* ((x (- i m)) (g (+ (* a x x) (* 2 b x) c)))
(if (< (ilog 2 g) (vector-ref sieve i))
(let* ((ys (smooth g fb)) (rel (make-rel x ys)))
(if (pair? ys)
(if (<= (car ys) f)
(set! rels (cons rel rels))
(set! parts (cons rel parts))))))))
(values rels parts)))
(define (match parts)
(define (lt? a b) (< (cadr a) (cadr b)))
(let loop ((parts (sort lt? parts)) (prev (list 0 0)) (zs (list)))
(cond ((null? parts) zs)
((= (cadar parts) (cadr prev))
(loop (cdr parts) prev
(cons (cons (* (caar parts) (car prev))
(merge > (cdar parts) (cdr prev))) zs)))
(else (loop (cdr parts) (car parts) zs)))))
(define (qs n f m)
(define (make-odd q) (if (odd? q) q (+ q 1)))
(call-with-values
(lambda () (factor-base n f))
(lambda (fb ts ls e len-fb)
(when verbose? (display "Factor base of ")
(display len-fb) (display " primes.") (newline))
(let loop ((q (make-odd (isqrt (quotient (isqrt (+ n n)) m)))) (rels (list)) (parts (list)))
(if (not (and (prime? q) (= (jacobi n q) 1))) (loop (+ q 2) rels parts)
(let* ((a (* q q)) (b (apply min (lift-root n q))) (q-inv (inverse q n)))
(when verbose? (display "q=") (display q) (display ": "))
(call-with-values
(lambda () (sieve n fb ts ls e f m a b q-inv))
(lambda (rs ps)
(let* ((rels (append rs rels)) (parts (append ps parts))
(matches (match parts)) (len-rels (length rels))
(len-parts (length parts)) (len-matches (length matches)))
(when verbose? (display len-rels) (display " smooths, ")
(display len-parts) (display " partials, ")
(display len-matches) (display " matches.") (newline))
(if (< (+ len-rels len-matches -10) len-fb) (loop (+ q 2) rels parts)
(let ((fact (solve n f fb (append rels matches))))
(if fact fact (loop (+ q 2) rels parts)))))))))))))
(define (make-expo-vector f fb rel)
(define (add-1bit x) (if (zero? x) 1 0))
(let loop ((fb fb) (rel rel) (prev -2) (es (list)))
(cond ((and (null? fb) (null? rel)) (list->vector (reverse es)))
((null? fb) (loop fb (cdr rel) prev (cons (add-1bit (car es)) (cdr es))))
((null? rel) (loop (cdr fb) rel prev (cons 0 es)))
((< f (car rel)) (loop fb (cdr rel) prev 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 rel) (cons 1 es)))
(else (loop (cdr fb) rel prev (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 (left-most-one vec r) ; column of left-most 1, or -1 if all zero
(let* ((row (vector-ref vec r)) (len (vector-length row)))
(let loop ((i 0))
(cond ((= i len) -1)
((= (vector-ref row i) 1) i)
(else (loop (+ i 1)))))))
(define (pivot-row expo c)
(let ((max-r (vector-length expo)))
(let loop ((r 0))
(if (= r max-r) r
(if (= (left-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)
(define (root y2)
(let loop ((y2 (sort < y2)) (s 1))
(if (null? y2) s
(loop (cddr y2) (* s (car y2))))))
(let* ((h (vector-ref hist r)) (h-len (vector-length h)))
(let loop ((i 0) (x 1) (y2 (list)))
(cond ((= i h-len)
(let ((g (gcd (- x (root y2)) n)))
(if (< 1 g n) g #f)))
((= (vector-ref h i) 1)
(loop (+ i 1) (* x (car (vector-ref rels i)))
(append (cdr (vector-ref rels i)) y2)))
(else (loop (+ i 1) x y2))))))
(define (solve n f fb rels)
(let* ((fb (reverse (cons -1 fb))) (fb-len (length fb)) (rel-len (length rels))
(expo (list->vector (map (lambda (rel) (make-expo-vector f fb (cdr rel))) rels)))
(hist (make-identity-matrix rel-len))
(rels (list->vector rels)))
(do ((c 0 (+ c 1))) ((= c fb-len))
(let ((p (pivot-row expo c)))
(do ((r (+ p 1) (+ r 1))) ((<= rel-len r))
(when (= (left-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)))))))
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