; rsa cryptography
(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 (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 (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 (euclid x y)
(let loop ((a 1) (b 0) (g x) (u 0) (v 1) (w y))
(if (zero? w) (values a b g)
(let ((q (quotient g w)))
(loop u v w (- a (* q u)) (- b (* q v)) (- g (* q w)))))))
(define (inverse x m)
(if (not (= (gcd x m) 1))
(error 'inverse "divisor must be coprime to modulus")
(call-with-values
(lambda () (euclid x m))
(lambda (a b g) (modulo a m)))))
(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 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 (prime? n)
(define (miller? 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 (chain m f g x0 x1)
(let loop ((ms (digits m 2)) (u x0) (v x1))
(cond ((null? ms) (values u v))
((zero? (car ms)) (loop (cdr ms) (f u) (g u v)))
(else (loop (cdr ms) (g u v) (f v))))))
(define (lucas? n)
(let loop ((a 11) (b 7))
(let ((d (- (* a a) (* 4 b))))
(cond ((square? d) (loop (+ a 2) (+ b 1)))
((not (= (gcd n (* 2 a b d)) 1))
(loop (+ a 2) (+ b 2)))
(else (let* ((x1 (modulo (- (* a a (inverse b n)) 2) n))
(m (quotient (- n (jacobi d n)) 2))
(f (lambda (u) (modulo (- (* u u) 2) n)))
(g (lambda (u v) (modulo (- (* u v) x1) n))))
(let-values (((xm xm1) (chain m f g 2 x1)))
(zero? (modulo (- (* x1 xm) (* 2 xm1)) n)))))))))
(cond ((not (and (integer? n) (positive? n)))
(error 'prime? "must be positive integer"))
((= n 1) #f) ((even? n) (= n 2)) ((zero? (modulo n 3)) (= n 3))
(else (and (miller? n 2) (miller? n 3) (lucas? n)))))
(define (keygen k e)
(define (gen k)
(let loop ((v (randint (expt 2 (- k 1)) (expt 2 k))))
(if (or (< 1 (gcd e (- v 1)))
(not (= (modulo v 4) 3))
(not (prime? v)))
(loop (+ v 1))
v)))
(let* ((k2 (quotient k 2)) (p (gen k2)) (q (gen k2))
(d (inverse e (* (- p 1) (- q 1)))))
(values (* p q) d)))
(define (crypt text modulus key)
(expm text key modulus))
(define n #f)
(define d #f)
(define e 65537)
(let-values (((nn dd) (keygen 32 e)))
(set! n nn) (set! d dd))
(display n) (newline)
(display d) (newline)
(define c (crypt 42 n e))
(display c) (newline)
(display (crypt c n d)) (newline)
(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 (factors n)
(if (even? n) (cons 2 (factors (/ n 2)))
(let ((s (+ (isqrt n) 1)))
(let loop ((u (+ s s 1)) (v 1) (r (- (* s s) n)))
(cond ((positive? r) (loop u (+ v 2) (- r v)))
((negative? r) (loop (+ u 2) v (+ r u)))
((= (- u v) 2) (list (/ (+ u v -2) 2)))
(else (append (factors (/ (+ u v -2) 2))
(factors (/ (- u v) 2)))))))))
(display (factors n)) (newline)