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.
Programming Praxis 
[…] 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 [/sourcecode]
#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))I’m slowly working my way through old exercises now that I have some free time. Here’s
my attempt in Common Lisp; I’m trying to learn the language, even though I already had Pythoned Miller-Rabin previously.
Wow, so much code needed in OCaml to solve this exercise in a self-contained fashion! First of all, a general-purpose function to return a random
Big_intbetween 0 and a specifiedmax(exclusive):let random_big_int = let open Big_int in let open Nat in let random_limb = match length_of_digit with | 64 -> fun nat ofs tmp -> set_digit_nat nat ofs (Random.bits ()); shift_left_nat nat ofs 1 tmp 0 30; set_digit_nat tmp 0 (Random.bits ()); lor_digit_nat nat ofs tmp 0; shift_left_nat nat ofs 1 tmp 0 30; set_digit_nat tmp 0 (Random.bits () land 7); lor_digit_nat nat ofs tmp 0 | 32 -> fun nat ofs tmp -> set_digit_nat nat ofs (Random.bits ()); shift_left_nat nat ofs 1 tmp 0 30; set_digit_nat tmp 0 (Random.bits () land 3); lor_digit_nat nat ofs tmp 0 | _ -> assert false in fun max -> let nat = nat_of_big_int max in let len = num_digits_nat nat 0 (length_nat nat) in let res = create_nat len and tmp = create_nat 1 in for i = 0 to len - 1 do random_limb res i tmp done; mod_big_int (big_int_of_nat res) max(You can’t go much lower level than this.) Next, some utility functions on
Big_ints:let is_zero_big_int n = Big_int.sign_big_int n == 0 let is_even_big_int n = Big_int.( is_zero_big_int (and_big_int n unit_big_int) ) let modsquare_big_int x n = Big_int.( mod_big_int (square_big_int x) n) let modexp_big_int x e n = let open Big_int in let rec go y z e = if is_zero_big_int e then y else let y = if is_even_big_int e then y else mod_big_int (mult_big_int y z) n in go y (modsquare_big_int z n) (shift_right_big_int e 1) in go unit_big_int x eFinally, an imperative-style Rabin-Miller test:
let is_prime n = let open Big_int in if le_big_int n unit_big_int || is_even_big_int n then invalid_arg "is_prime" else let m = pred_big_int n in let r = ref 0 and s = ref m in while is_even_big_int !s do incr r; s := shift_right_big_int !s 1 done; try for i = 1 to 50 do let a = add_int_big_int 2 (random_big_int (add_int_big_int (-2) n)) in let x = ref (modexp_big_int a !s n) and j = ref !r in let any = ref (eq_big_int !x unit_big_int) in while !j != 0 && not !any do if eq_big_int !x m then any := true else x := modsquare_big_int !x n; decr j done; if not !any then raise Exit done; true with Exit -> falseTwo optimizations were applied: first, the random a should be greater than 1; second, the successive powers of a in the inner loop are calculated inductively by (modular) squaring.
I made a mistake in
random_big_int. Lines 10 and 11 are wrong, they should read:shift_left_nat nat ofs 1 tmp 0 4; set_digit_nat tmp 0 (Random.bits () land 15);This is a Haskell program I wrote (ironically enough around May 2009, though I hadn’t heard about this website back then.) This was my first relatively non-trivial Haskell program.
import System.Random import System (getArgs) -- return number N as (s,d) where N = (2^S)*d pow2factor n | odd n = (0, n) | even n = (s+1, d) where (s,d) = pow2factor (n `div` 2) -- compute (a^n) (mod m) -- (efficient algorithm, doing modulo arithmetic after each step) square x = x*x pow a 0 m = 1 pow a n m | odd n = (a * (pow a (n - 1) m)) `mod` m | even n = (square (pow a (n `div` 2) m)) `mod` m -- Miller-Rabin: given an integer n, determine if it is prime with -- probability 4 ^ -k, where k is the number of tests performed. probablyPrime n tests = let (s, d) = pow2factor (n-1) in not (any (\x -> testComposite x s d) tests) where testComposite a s d = (pow a d n) `notElem` [1, n-1] && all (\r -> (pow a (r*d) n) /= n-1) (take (s-1) (iterate (2*) 2)) -- main testPrime n = do rnd <- newStdGen return (probablyPrime n (take 10 (randomRs (2, n-2) rnd))) genprime :: Integer -> IO Integer genprime bits = do rnd <- getStdGen thePrime <- firstprime (randomRs (2, 2^(bits-1)) rnd) return thePrime where firstprime (h:t) = let candidate = 2*h+1 in do isPrime <- testPrime candidate if isPrime then return candidate else firstprime t main = do args <- getArgs case args of ["-g", bits] -> do x <- genprime ((read bits) :: Integer) putStrLn (show x) ["-t", n] -> do isPrime <- testPrime ((read n) :: Integer) putStrLn (if isPrime then "prime" else "composite") otherwise -> error "Usage: miller-rabin [-g bits] [-t number]"To test 2^89 -1 use,
.\miller-rabin.exe -t 618970019642690137449562111
Rewrote the Haskell Miller-Rabin in Clojure as my first Clojure program. It is line-by-line translation, pretty much. Main difference is that one can’t generate random numbers > 2^63, so that is taken into account when generating tests.
(defn factor-2s "return argument N as vector [s,d] where N = (2^S)*d" [n] (loop [n' n e 0] (if (even? n') (recur (/ n' 2) (inc e)) [e n']))) (defn square [x] (* x x)) (defn pow "compute (a^n) (mod m) (efficient algorithm, doing modulo arithmetic after each step)" [a n m] (cond (zero? n) 1 (odd? n) (mod (* a (pow a (dec n) m)) m) (even? n) (mod (square (pow a (/ n 2) m)) m))) (defn random-gen "Lazily computed list of random numbers" [n] (map #(long (rand %1)) (cycle [n]))) (defn probably-prime "Miller-Rabin: given an integer n, determine if it is prime with probability 1 - (4^-k), where k is the number of tests performed." [accuracy n] (let [[s d] (factor-2s (dec n)) composite-witness? (fn [a] (and (let [t (pow a d n)] (and (not= t 1) (not= t (dec n)))) (every? (fn [r] (not= (pow a (* r d) n) (dec n))) (take (dec s) (iterate #(+ %1 %1) 2)))))] (cond (< n 2) false (= n 2) true (even? n) false true (not-any? composite-witness? (take accuracy (map (partial + 2) (random-gen (min (- n 4) 0x7FFFFFFFFFFFFFFF)))))))) (def prime-accuracy 10) (def prime? (partial probably-prime prime-accuracy)){
{前几天|最近|这段时间|这些天|前段时间|前些日子|没多久前|不久前}刚开始{碰|啃|练习|接触|学|接触到|学习} python，现在{碰到|遇到|遭遇到|接触到|要用到}验证码问题，{网上|上网}{查了下|google了一下|百度了一下|谷歌了一下|baidu了一下|百度了一会|谷歌了一会|google了一会|搜了一下|搜了一会}{相关|}{内容|知识|资料|教程}，{知道|得知|获悉|了解到|了解|好像|似乎}用python获取验证码{的人|}还是{比较多的|不少|不在少数|挺多的|}。自己琢磨了{一番|一下|一下午|一整天|一早上|一晚上|整个上午|整个下午|整个晚上}，用的是{开源的|免费的|不用钱的|不用银子的|不用付费的}打码工具tesseractor，但{始终|最后|最终|后来|至今|到现在|到目前}都没{搞出来|搞定|成功|获取验证码|成功获取验证码|获取到验证码|弄出验证码|解决验证码问题|搞出验证码}，{不知道|不清楚|不了解|不明白|没弄懂|没明白|没了解|没搞清|没搞懂|没搞明白}是什么{问题|原因|情况}，难道是{免费没好货|免费的问题|天下没有免费的午餐}？{有没有|有无|知不知|知不知道|有人知道|有没人知道|有没谁知道|有谁知道|有谁了解|有谁用过}{什么|哪个|哪些|哪一些|哪个}付费的{推荐|软件|平台|应用|工具}吗？
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{前几天|最近|这段时间|这些天|前段时间|前些日子|没多久前|不久前}刚开始{碰|啃|练习|接触|学|接触到|学习}
python，现在{碰到|遇到|遭遇到|接触到|要用到}验证码问题，{通过|测试了下|弄了下|搞了下}本地可以{解析|获取|弄|搞|读取|读}出{一个|}字符串，{然后|但是|可是|接着|不过}再提交{的时候|后|之后|过后|以后}这个验证码已经{变|改变|变化}了，{我|}{在这里|在这|现在|如今}想{问|咨询|询问|了解|搞明白|弄懂}的是{如何|怎样|怎么}{才|}能{保证|保持}这两{者|个}{一致|相同|不变|}。