## 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)))))))