; generating large random primes

(define (drop-while pred? xs)
  (let loop ((xs xs))
    (if (or (null? xs) (not (pred? (car xs)))) xs
      (loop (cdr xs)))))

(define (fortune xs)
  (let loop ((n 1) (x #f) (xs xs))
    (cond ((null? xs) x)
          ((< (rand) (/ n))
            (loop (+ n 1) (car xs) (cdr xs)))
          (else (loop (+ n 1) x (cdr xs))))))

(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 rand
  (let ((seed 0.3141592654))
    (lambda args
      (set! seed
        (if (pair? args)
            (sin (car args))
            (let ((x (* seed 147.0)))
              (- x (floor x)))))
      seed)))

(define (randint first past)
  (inexact->exact (floor (+ (* (- past first) (rand)) first))))

(define (primes n) ; list of primes not exceeding n
  (let* ((len (quotient (- n 1) 2)) (bits (make-vector len #t)))
    (let loop ((i 0) (p 3) (ps (list 2)))
      (cond ((< n (* p p))
              (do ((i i (+ i 1)) (p p (+ p 2))
                   (ps ps (if (vector-ref bits i) (cons p ps) ps)))
                  ((= i len) (reverse ps))))
            ((vector-ref bits i)
              (do ((j (+ (* 2 i i) (* 6 i) 3) (+ j p)))
                  ((<= len j) (loop (+ i 1) (+ p 2) (cons p ps)))
                (vector-set! bits j #f)))
            (else (loop (+ i 1) (+ p 2) ps))))))

(define prime? ; strong pseudoprime to prime bases less than 100
  (let* ((ps (primes 100)) (p100 (apply * ps)))
    (lambda (n)
      (define (expm b e m)
        (let loop ((b b) (e e) (x 1))
          (if (zero? e) x
            (loop (modulo (* b b) m) (quotient e 2)
                  (if (odd? e) (modulo (* b x) m) x)))))
      (define (spsp? n a) ; #t if n is a strong pseudoprime base a
        (do ((d (- n 1) (/ d 2)) (s 0 (+ s 1)))
            ((odd? d) (if (= (expm a d n) 1) #t
              (do ((r 0 (+ r 1)))
                  ((or (= (expm a (* d (expt 2 r)) n) (- n 1)) (= r s))
                    (< r s)))))))
      (if (< n 2) #f (if (< 1 (gcd n p100)) (if (member n ps) #t #f)
        (do ((ps ps (cdr ps)))
            ((or (null? ps) (not (spsp? n (car ps)))) (null? ps))))))))

(define (pocklington p k a)
  (and (odd? p) (prime? p)
       (< 1 k (* 2 (+ p 1)))
       (positive? (modulo k p))
       (let ((n (+ (* 2 k p) 1)))
         (and (< 1 a n)
              (= (expm a (/ (- n 1) 2) n) (- n 1))
              (= (gcd (+ (expm a k n) 1) n) 1)
              n)))) ; return n if prime, else #f

(define (ratchet p lo hi verbose?)
  (define (pock p k a n)
    (and (= (expm a (/ (- n 1) 2) n) (- n 1))
         (= (gcd (+ (expm a k n) 1) n) 1)))
  (let loop ((k (randint lo hi)))
    (let ((n (+ (* 2 k p) 1)))
      (cond ((pock p k 2 n) (when verbose? (display (list p k 2)) (newline)) n)
            ((pock p k 3 n) (when verbose? (display (list p k 3)) (newline)) n)
            (else (loop (randint lo hi)))))))

(define (rand-prime lo hi . args)
  (let ((verbose? (if (pair? args) (car args) #f)))
    (if (< hi 10000) (fortune (drop-while (lambda (x) (< x lo)) (primes hi)))
      (let loop ((p (fortune (cdr (primes 10000)))))
        (if (<= lo p) p
          (let ((k-lo (ceiling (/ (- lo 1) 2 p)))
                (k-hi (floor (/ (- hi 1) 2 p))))
            (loop (ratchet p (if (< (* 2 p) k-lo) 1 k-lo)
                           (min (* 2 p) k-hi) verbose?))))))))

(display (pocklington 2333 2001 2)) (newline)
(display (pocklington 2333 2017 2)) (newline)

(display (rand-prime #e1e49 #e1e50 #t)) (newline)