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; frobenius pseudoprimality test (define rand #f) (define randint #f) (let ((two31 #x80000000) (a (make-vector 56 -1)) (fptr #f)) (define (mod-diff x y) (modulo (- x y) two31)) ; generic version ; (define (mod-diff x y) (logand (- x y) #x7FFFFFFF)) ; fast version (define (flip-cycle) (do ((ii 1 (+ ii 1)) (jj 32 (+ jj 1))) ((< 55 jj)) (vector-set! a ii (mod-diff (vector-ref a ii) (vector-ref a jj)))) (do ((ii 25 (+ ii 1)) (jj 1 (+ jj 1))) ((< 55 ii)) (vector-set! a ii (mod-diff (vector-ref a ii) (vector-ref a jj)))) (set! fptr 54) (vector-ref a 55)) (define (init-rand seed) (let* ((seed (mod-diff seed 0)) (prev seed) (next 1)) (vector-set! a 55 prev) (do ((i 21 (modulo (+ i 21) 55))) ((zero? i)) (vector-set! a i next) (set! next (mod-diff prev next)) (set! seed (+ (quotient seed 2) (if (odd? seed) #x40000000 0))) (set! next (mod-diff next seed)) (set! prev (vector-ref a i))) (flip-cycle) (flip-cycle) (flip-cycle) (flip-cycle) (flip-cycle))) (define (next-rand) (if (negative? (vector-ref a fptr)) (flip-cycle) (let ((next (vector-ref a fptr))) (set! fptr (- fptr 1)) next))) (define (unif-rand m) (let ((t (- two31 (modulo two31 m)))) (let loop ((r (next-rand))) (if (<= t r) (loop (next-rand)) (modulo r m))))) (init-rand 19380110) ; happy birthday donald e knuth (set! rand (lambda seed (cond ((null? seed) (/ (next-rand) two31)) ((eq? (car seed) 'get) (cons fptr (vector->list a))) ((eq? (car seed) 'set) (set! fptr (caadr seed)) (set! a (list->vector (cdadr seed)))) (else (/ (init-rand (modulo (numerator (inexact->exact (car seed))) two31)) two31))))) (set! randint (lambda args (cond ((null? (cdr args)) (if (< (car args) two31) (unif-rand (car args)) (floor (* (next-rand) (car args))))) ((< (car args) (cadr args)) (let ((span (- (cadr args) (car args)))) (+ (car args) (if (< span two31) (unif-rand span) (floor (* (next-rand) span)))))) (else (let ((span (- (car args) (cadr args)))) (- (car args) (if (< span two31) (unif-rand span) (floor (* (next-rand) span)))))))))) (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 (digits n . args) (let ((b (if (null? args) 10 (car args)))) (let loop ((n n) (d '())) (if (zero? n) d (loop (quotient n b) (cons (modulo n b) d)))))) (define square? (let ((q11 (make-vector 11 #f)) (q63 (make-vector 63 #f)) (q64 (make-vector 64 #f)) (q65 (make-vector 65 #f))) (do ((k 0 (+ k 1))) ((< 5 k)) (vector-set! q11 (modulo (* k k) 11) #t)) (do ((k 0 (+ k 1))) ((< 31 k)) (vector-set! q63 (modulo (* k k) 63) #t)) (do ((k 0 (+ k 1))) ((< 31 k)) (vector-set! q64 (modulo (* k k) 64) #t)) (do ((k 0 (+ k 1))) ((< 32 k)) (vector-set! q65 (modulo (* k k) 65) #t)) (lambda (n) (if (not (integer? n)) (error 'square? "must be integer") (if (< n 1) #f (if (not (vector-ref q64 (modulo n 64))) #f (let ((r (modulo n 45045))) (if (not (vector-ref q63 (modulo r 63))) #f (if (not (vector-ref q65 (modulo r 65))) #f (if (not (vector-ref q11 (modulo r 11))) #f (let ((q (isqrt n))) (if (= (* q q) n) q #f)))))))))))) (define (jacobi a n) (if (not (and (integer? a) (integer? n) (positive? n) (odd? n))) (error 'jacobi "modulus must be positive odd integer") (let jacobi ((a a) (n n)) (cond ((= a 0) 0) ((= a 1) 1) ((= a 2) (case (modulo n 8) ((1 7) 1) ((3 5) -1))) ((even? a) (* (jacobi 2 n) (jacobi (quotient a 2) n))) ((< n a) (jacobi (modulo a n) n)) ((and (= (modulo a 4) 3) (= (modulo n 4) 3)) (- (jacobi n a))) (else (jacobi n a)))))) (define (inverse x m) (let loop ((x x) (b m) (a 0) (u 1)) (if (zero? x) (if (= b 1) (modulo a m) (error 'inverse "must be coprime")) (let* ((q (quotient b x))) (loop (modulo b x) x u (- a (* u q))))))) (define (frobenius-pseudoprime? n) (let loop ((a (randint 1 n)) (b (randint 1 n))) (display "a = ") (display a) (newline) (display "b = ") (display b) (newline) (let ((d (- (* a a) (* 4 b)))) (display "d = ") (display d) (newline) (if (or (= a b) (square? d) (not (= (gcd (* 2 a b d) n) 1))) (loop (randint 1 n) (randint 1 n)) (let* ((m (/ (- n (jacobi d n)) 2)) (w (modulo (- (* a a (inverse b n)) 2) n))) (display "m = ") (display m) (newline) (display "w = ") (display w) (newline) (let loop ((u 2) (v w) (ms (digits m 2))) (display "u, v = ") (display u) (display " ") (display v) (newline) (if (pair? ms) (if (zero? (car ms)) (loop (modulo (- (* u u) 2) n) (modulo (- (* u v) w) n) (cdr ms)) (loop (modulo (- (* u v) w) n) (modulo (- (* v v) 2) n) (cdr ms))) (if (positive? (modulo (- (* u w) (* 2 v)) n)) #f (let ((x (expm b (/ (- n 1) 2) n))) (zero? (modulo (- (* u x) 2) n))))))))))) (define (strong-pseudoprime? n a) (let loop ((r 0) (s (- n 1))) (if (even? s) (loop (+ r 1) (/ s 2)) (if (= (expm a s n) 1) #t (let loop ((r r) (s s)) (cond ((zero? r) #f) ((= (expm a s n) (- n 1)) #t) (else (loop (- r 1) (* s 2))))))))) (define (prime? n) (let loop ((ps (list 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97))) (if (pair? n) (if (zero? (modulo n (car ps))) (= n (car ps)) (loop (cdr ps))) (and (strong-pseudoprime? n 2) (strong-pseudoprime? n 3) (frobenius-pseudoprime? n))))) (display (prime? 3825123056546413051)) (newline) (display (prime? (- (expt 2 89) 1))) (newline)
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