    -- |
    -- Module      : Crypto.Number.Prime
    -- License     : BSD-style
    -- Maintainer  : Vincent Hanquez <vincent@snarc.org>
    -- Stability   : experimental
    -- Portability : Good

    {-# LANGUAGE BangPatterns #-}
    module Crypto.Number.Prime
        (
          generatePrime
        , generateSafePrime
        , isProbablyPrime
        , findPrimeFrom
        , findPrimeFromWith
        , primalityTestMillerRabin
        , primalityTestNaive
        , primalityTestFermat
        , isCoprime
        ) where

    import Crypto.Internal.Imports

    import Crypto.Number.Compat
    import Crypto.Number.Generate
    import Crypto.Number.Basic (sqrti, gcde)
    import Crypto.Number.ModArithmetic (expSafe)
    import Crypto.Random.Types
    import Crypto.Random.Probabilistic

    import Data.Bits

    -- | 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 :: Integer -> Bool
    isProbablyPrime !n
        | any (\p -> p `divides` n) (filter (< n) firstPrimes) = False
        | primalityTestFermat 50 (n`div`2) n                   = primalityTestMillerRabin 30 n
        | otherwise                                            = False

    -- | generate a prime number of the required bitsize
    generatePrime :: MonadRandom m => Int -> m Integer
    generatePrime bits = do
        sp <- generateParams bits (Just SetTwoHighest) True
        return $ findPrimeFrom 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 :: MonadRandom m => Int -> m Integer
    generateSafePrime bits = do
        sp <- generateParams bits (Just SetTwoHighest) True
        let p = findPrimeFromWith (\i -> isProbablyPrime (2*i+1)) (sp `div` 2)
        return (2*p+1)

    -- | find a prime from a starting point where the property hold.
    findPrimeFromWith :: (Integer -> Bool) -> Integer -> Integer
    findPrimeFromWith prop !n
        | even n        = findPrimeFromWith prop (n+1)
        | otherwise     =
            if not (isProbablyPrime n)
                then findPrimeFromWith prop (n+2)
                else
                    if prop n
                        then n
                        else findPrimeFromWith prop (n+2)

    -- | find a prime from a starting point with no specific property.
    findPrimeFrom :: Integer -> Integer
    findPrimeFrom n =
        case gmpNextPrime n of
            GmpSupported p -> p
            GmpUnsupported -> findPrimeFromWith (\_ -> True) 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 :: Int -> Integer -> Bool
    primalityTestMillerRabin tries !n =
        case gmpTestPrimeMillerRabin tries n of
            GmpSupported b -> b
            GmpUnsupported -> probabilistic run
      where
        run
            | n <= 3     = error "Miller-Rabin requires tested value to be > 3"
            | even n     = return False
            | tries <= 0 = error "Miller-Rabin tries need to be > 0"
            | otherwise  = loop <$> generateTries tries

        !nm1 = n-1
        !nm2 = n-2

        (!s,!d) = (factorise 0 nm1)

        generateTries 0 = return []
        generateTries t = do
            v  <- generateBetween 2 nm2
            vs <- generateTries (t-1)
            return (v:vs)

        -- 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 = expSafe

        -- 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 = expSafe 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 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