{-# LANGUAGE BangPatterns #-} -- | -- Module : Crypto.Number.Prime -- License : BSD-style -- Maintainer : Vincent Hanquez -- Stability : experimental -- Portability : Good module Crypto.Number.Prime ( generatePrime , generateSafePrime , isProbablyPrime , findPrimeFrom , findPrimeFromWith , primalityTestNaive , primalityTestMillerRabin , primalityTestFermat , isCoprime ) where import Crypto.Random.API import Data.Bits import Crypto.Number.Generate import Crypto.Number.Basic (sqrti, gcde_binary) import Crypto.Number.ModArithmetic (exponantiation) -- | returns if the number is probably prime. -- first a list of small primes are implicitely tested for divisibility, -- then a fermat primality test is used with arbitrary numbers and -- then the Miller Rabin algorithm is used with an accuracy of 30 recursions isProbablyPrime :: CPRG g => g -> Integer -> (Bool, g) isProbablyPrime rng !n | any (\p -> p `divides` n) (filter (< n) firstPrimes) = (False, rng) | primalityTestFermat 50 (n`div`2) n = primalityTestMillerRabin rng 30 n | otherwise = (False, rng) -- | generate a prime number of the required bitsize generatePrime :: CPRG g => g -> Int -> (Integer, g) generatePrime rng bits = let (sp, rng') = generateOfSize rng bits in findPrimeFrom rng' sp -- | generate a prime number of the form 2p+1 where p is also prime. -- it is also knowed as a Sophie Germaine prime or safe prime. -- -- The number of safe prime is significantly smaller to the number of prime, -- as such it shouldn't be used if this number is supposed to be kept safe. generateSafePrime :: CPRG g => g -> Int -> (Integer, g) generateSafePrime rng bits = let (sp, rng') = generateOfSize rng bits (p, rng'') = findPrimeFromWith rng' (\g i -> isProbablyPrime g (2*i+1)) (sp `div` 2) in (2*p+1, rng'') -- | find a prime from a starting point where the property hold. findPrimeFromWith :: CPRG g => g -> (g -> Integer -> (Bool,g)) -> Integer -> (Integer, g) findPrimeFromWith rng prop !n | even n = findPrimeFromWith rng prop (n+1) | otherwise = case isProbablyPrime rng n of (False, rng') -> findPrimeFromWith rng' prop (n+2) (True, rng') -> case prop rng' n of (False, rng'') -> findPrimeFromWith rng'' prop (n+2) (True, rng'') -> (n, rng'') -- | find a prime from a starting point with no specific property. findPrimeFrom :: CPRG g => g -> Integer -> (Integer, g) findPrimeFrom rng n = findPrimeFromWith rng (\g _ -> (True, g)) n -- | Miller Rabin algorithm return if the number is probably prime or composite. -- the tries parameter is the number of recursion, that determines the accuracy of the test. primalityTestMillerRabin :: CPRG g => g -> Int -> Integer -> (Bool, g) primalityTestMillerRabin rng tries !n | n <= 3 = error "Miller-Rabin requires tested value to be > 3" | even n = (False, rng) | tries <= 0 = error "Miller-Rabin tries need to be > 0" | otherwise = let (witnesses, rng') = generateTries tries rng in (loop witnesses, rng') where !nm1 = n-1 !nm2 = n-2 (!s,!d) = (factorise 0 nm1) generateTries 0 g = ([], g) generateTries t g = let (v,g') = generateBetween g 2 nm2 (vs,g'') = generateTries (t-1) g' in (v:vs, g'') -- factorise n-1 into the form 2^s*d factorise :: Integer -> Integer -> (Integer, Integer) factorise !si !vi | vi `testBit` 0 = (si, vi) | otherwise = factorise (si+1) (vi `shiftR` 1) -- probably faster to not shift v continously, but just once. expmod = exponantiation -- when iteration reach zero, we have a probable prime loop [] = True loop (w:ws) = let x = expmod w d n in if x == (1 :: Integer) || x == nm1 then loop ws else loop' ws ((x*x) `mod` n) 1 -- loop from 1 to s-1. if we reach the end then it's composite loop' ws !x2 !r | r == s = False | x2 == 1 = False | x2 /= nm1 = loop' ws ((x2*x2) `mod` n) (r+1) | otherwise = loop ws {- n < z -> witness to test 1373653 [2,3] 9080191 [31,73] 4759123141 [2,7,61] 2152302898747 [2,3,5,7,11] 3474749660383 [2,3,5,7,11,13] 341550071728321 [2,3,5,7,11,13,17] -} -- | Probabilitic Test using Fermat primility test. -- Beware of Carmichael numbers that are Fermat liars, i.e. this test -- is useless for them. always combines with some other test. primalityTestFermat :: Int -- ^ number of iterations of the algorithm -> Integer -- ^ starting a -> Integer -- ^ number to test for primality -> Bool primalityTestFermat n a p = and $ map expTest [a..(a+fromIntegral n)] where !pm1 = p-1 expTest i = exponantiation i pm1 p == 1 -- | Test naively is integer is prime. -- while naive, we skip even number and stop iteration at i > sqrt(n) primalityTestNaive :: Integer -> Bool primalityTestNaive n | n <= 1 = False | n == 2 = True | even n = False | otherwise = search 3 where !ubound = snd $ sqrti n search !i | i > ubound = True | i `divides` n = False | otherwise = search (i+2) -- | Test is two integer are coprime to each other isCoprime :: Integer -> Integer -> Bool isCoprime m n = case gcde_binary m n of (_,_,d) -> d == 1 -- | list of the first primes till 2903.. firstPrimes :: [Integer] firstPrimes = [ 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 , 101 , 103 , 107 , 109 , 113 , 127 , 131 , 137 , 139 , 149 , 151 , 157 , 163 , 167 , 173 , 179 , 181 , 191 , 193 , 197 , 199 , 211 , 223 , 227 , 229 , 233 , 239 , 241 , 251 , 257 , 263 , 269 , 271 , 277 , 281 , 283 , 293 , 307 , 311 , 313 , 317 , 331 , 337 , 347 , 349 , 353 , 359 , 367 , 373 , 379 , 383 , 389 , 397 , 401 , 409 , 419 , 421 , 431 , 433 , 439 , 443 , 449 , 457 , 461 , 463 , 467 , 479 , 487 , 491 , 499 , 503 , 509 , 521 , 523 , 541 , 547 , 557 , 563 , 569 , 571 , 577 , 587 , 593 , 599 , 601 , 607 , 613 , 617 , 619 , 631 , 641 , 643 , 647 , 653 , 659 , 661 , 673 , 677 , 683 , 691 , 701 , 709 , 719 , 727 , 733 , 739 , 743 , 751 , 757 , 761 , 769 , 773 , 787 , 797 , 809 , 811 , 821 , 823 , 827 , 829 , 839 , 853 , 857 , 859 , 863 , 877 , 881 , 883 , 887 , 907 , 911 , 919 , 929 , 937 , 941 , 947 , 953 , 967 , 971 , 977 , 983 , 991 , 997 , 1009 , 1013 , 1019 , 1021 , 1031 , 1033 , 1039 , 1049 , 1051 , 1061 , 1063 , 1069 , 1087 , 1091 , 1093 , 1097 , 1103 , 1109 , 1117 , 1123 , 1129 , 1151 , 1153 , 1163 , 1171 , 1181 , 1187 , 1193 , 1201 , 1213 , 1217 , 1223 , 1229 , 1231 , 1237 , 1249 , 1259 , 1277 , 1279 , 1283 , 1289 , 1291 , 1297 , 1301 , 1303 , 1307 , 1319 , 1321 , 1327 , 1361 , 1367 , 1373 , 1381 , 1399 , 1409 , 1423 , 1427 , 1429 , 1433 , 1439 , 1447 , 1451 , 1453 , 1459 , 1471 , 1481 , 1483 , 1487 , 1489 , 1493 , 1499 , 1511 , 1523 , 1531 , 1543 , 1549 , 1553 , 1559 , 1567 , 1571 , 1579 , 1583 , 1597 , 1601 , 1607 , 1609 , 1613 , 1619 , 1621 , 1627 , 1637 , 1657 , 1663 , 1667 , 1669 , 1693 , 1697 , 1699 , 1709 , 1721 , 1723 , 1733 , 1741 , 1747 , 1753 , 1759 , 1777 , 1783 , 1787 , 1789 , 1801 , 1811 , 1823 , 1831 , 1847 , 1861 , 1867 , 1871 , 1873 , 1877 , 1879 , 1889 , 1901 , 1907 , 1913 , 1931 , 1933 , 1949 , 1951 , 1973 , 1979 , 1987 , 1993 , 1997 , 1999 , 2003 , 2011 , 2017 , 2027 , 2029 , 2039 , 2053 , 2063 , 2069 , 2081 , 2083 , 2087 , 2089 , 2099 , 2111 , 2113 , 2129 , 2131 , 2137 , 2141 , 2143 , 2153 , 2161 , 2179 , 2203 , 2207 , 2213 , 2221 , 2237 , 2239 , 2243 , 2251 , 2267 , 2269 , 2273 , 2281 , 2287 , 2293 , 2297 , 2309 , 2311 , 2333 , 2339 , 2341 , 2347 , 2351 , 2357 , 2371 , 2377 , 2381 , 2383 , 2389 , 2393 , 2399 , 2411 , 2417 , 2423 , 2437 , 2441 , 2447 , 2459 , 2467 , 2473 , 2477 , 2503 , 2521 , 2531 , 2539 , 2543 , 2549 , 2551 , 2557 , 2579 , 2591 , 2593 , 2609 , 2617 , 2621 , 2633 , 2647 , 2657 , 2659 , 2663 , 2671 , 2677 , 2683 , 2687 , 2689 , 2693 , 2699 , 2707 , 2711 , 2713 , 2719 , 2729 , 2731 , 2741 , 2749 , 2753 , 2767 , 2777 , 2789 , 2791 , 2797 , 2801 , 2803 , 2819 , 2833 , 2837 , 2843 , 2851 , 2857 , 2861 , 2879 , 2887 , 2897 , 2903 ] {-# INLINE divides #-} divides :: Integer -> Integer -> Bool divides i n = n `mod` i == 0