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)))
    (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 ": "))
              (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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