Primality Checking
May 1, 2009
Although the math looks difficult, the code is quite straight forward. Check? calculates r and s in the outer loop, checks if as is 1 modulo n, and then the inner loop checks the other condition for each j from 0 to r-1.
(define (check? a n)
(let loop ((r 0) (s (- n 1)))
(if (even? s) (loop (+ r 1) (/ s 2))
(if (= (expm a s n) 1) #t
(let loop ((j 0) (s s))
(cond ((= j r) #f)
((= (expm a s n) (- n 1)) #t)
(else (loop (+ j 1) (* s 2)))))))))
Then prime? tests a few boundary conditions and performs fifty random calls to check?.
(define (prime? n)
(cond ((< n 2) #f) ((= n 2) #t) ((even? n) #f)
(else (let loop ((k 50))
(cond ((zero? k) #t)
((not (check? (randint 1 n) n)) #f)
(else (loop (- k 1))))))))
Expm (modular exponentiation) and randint come from the Standard Prelude. A quick test shows that 289-1 = 618970019642690137449562111 is prime:
> (prime? (- (expt 2 89) 1))
#t
You can run this code at http://codepad.org/rSDxFrZn.
Pages: 1 2
[...] Praxis – Primality Checking By Remco Niemeijer Today’s Programming Praxis problem is about checking whether or not a number is prime. We’re supposed [...]
My Haskell solution (see http://bonsaicode.wordpress.com/2009/05/01/programming-praxis-primality-checking/ for a version with comments):
import Control.Arrow import Data.Bits import Data.List import System.Random isPrime :: Integer -> StdGen -> Bool isPrime n g = let (s, d) = (length *** head) . span even $ iterate (flip div 2) (n - 1) xs = map (expm n d) . take 50 $ randomRs (2, n - 2) g in all (\x -> elem x [1, n - 1] || any (== n - 1) (take s $ iterate (expm n 2) x)) xs expm :: Integer -> Integer -> Integer -> Integer expm m e b = foldl' (\r (b', _) -> mod (r * b') m) 1 . filter (flip testBit 0 . snd) . zip (iterate (flip mod m . (^ 2)) b) $ takeWhile (> 0) $ iterate (flip shiftR 1) e main :: IO () main = print . isPrime (2 ^ 89 - 1) =<< getStdGen#lang scheme (require srfi/1 ; list-tabulate srfi/27) ; random-integer (define (factor2 n) ; return r and s s.t n = 2^r * s where s is odd (let loop ([r 0] [s n]) ; invariant: n = 2^r * s (let-values ([(q r) (quotient/remainder s 2)]) (if (zero? r) (loop (+ r 1) q) (values r s))))) (define (miller-rabin n) ; Input: n odd Output: n prime? (define (mod x) (modulo x n)) (define (^ x m) (cond [(zero? m) 1] [(even? m) (mod (sqr (^ x (/ m 2))))] [(odd? m) (mod (* x (^ x (- m 1))))])) (define (check? a) (let-values ([(r s) (factor2 (sub1 n))]) (and (member (^ a s) (list 1 (mod -1))) #t))) (andmap check? (list-tabulate 50 (λ (_) (+ 2 (random-integer (- n 3))))))) (define (prime? n) (cond [(< n 2) #f] [(= n 2) #t] [(even? n) #f] [else (miller-rabin n)])) (prime? (- (expt 2 89) 1))