diff --git a/Benchmarks/Benchmarks.hs b/Benchmarks/Benchmarks.hs new file mode 100644 --- /dev/null +++ b/Benchmarks/Benchmarks.hs @@ -0,0 +1,68 @@ +module Main where + +import Criterion.Main + +import Crypto.Number.Serialize +import Crypto.Number.Generate +import qualified Data.ByteString as B +import Crypto.Number.ModArithmetic +import Data.Bits + +primes = [3, 5, 7, 29, 31, 211, 2309, 2311, 30029, 200560490131, 304250263527209] +carmichaelNumbers = [41041, 62745, 63973, 75361, 101101, 126217, 172081, 188461, 278545, 340561] + +lg1, lg2 :: Integer +lg1 = 21389083291083903845902381390285907190274907230982112390820985903825329874812973821790321904790217490217409721904832974210974921740972109481490128430982190472109874802174907490271904124908210958093285098309582093850918902581290859012850829105809128590218590281905812905810928590128509128940821903829018390849839578967358920127598901248259797158249684571948075896458741905823982671490352896791052386357019528367902 +lg2 = 21392813098390824190840192812389082390812940821904891028439028490128904829104891208940835932882910839218309812093118249089871209347472901874902407219740921840928149087284397490128903843789289014374839281492038091283923091809734832974180398210938901284839274091749021709 + +bitsAndShift8 n i = (n `shiftR` i, n .&. 0xff) +modAndShift8 n i = (n `shiftR` i, n `mod` 0x100) + +main = defaultMain + [ bgroup "std ops" + [ bench "mod" $ nf (mod lg1) lg2 + , bench "rem" $ nf (rem lg1) lg2 + , bench "div" $ nf (div lg1) lg2 + , bench "quot" $ nf (quot lg1) lg2 + , bench "divmod" $ nf (divMod lg1) lg2 + , bench "quotRem" $ nf (quotRem lg1) lg2 + ] + , bgroup "divMod by 256" + [ bench "divmod 256" $ nf (divMod lg1) 256 + , bench "quotRem 256" $ nf (quotRem lg1) 256 + , bench "modAndShift 8" $ nf (modAndShift8 lg1) 8 + , bench "bitsAndShift 8" $ nf (bitsAndShift8 lg1) 8 + ] + , bgroup "serialization bs->i" + [ bench "8" $ nf os2ip b8 + , bench "32" $ nf os2ip b32 + , bench "64" $ nf os2ip b64 + , bench "256" $ nf os2ip b256 + , bench "1024" $ nf os2ip b1024 + ] + , bgroup "serialization i->bs" + [ bench "10" $ nf i2osp (2^10) + , bench "100" $ nf i2osp (2^100) + , bench "1000" $ nf i2osp (2^1000) + , bench "10000" $ nf i2osp (2^10000) + , bench "100000" $ nf i2osp (2^100000) + ] + , bgroup "serialization i->bs of size" + [ bench "10" $ nf (i2ospOf_ 4) (2^10) + , bench "100" $ nf (i2ospOf_ 16) (2^100) + , bench "1000" $ nf (i2ospOf_ 128) (2^1000) + , bench "10000" $ nf (i2ospOf_ 1560) (2^10000) + , bench "100000" $ nf (i2ospOf_ 12502) (2^100000) + ] + , bgroup "exponentiation" + [ bench "2^1234 mod 2^999" $ nf (exponantiation 2 1234) (2^999) + , bench "130^5432 mod 100^9990" $ nf (exponantiation 130 5432) (100^9999) + , bench "2^1234 mod 2^999" $ nf (exponantiation_rtl_binary 2 1234) (2^999) + , bench "130^5432 mod 100^9990" $ nf (exponantiation_rtl_binary 130 5432) (100^9999) + ] + ] + where b8 = B.replicate 8 0xf7 + b32 = B.replicate 32 0xf7 + b64 = B.replicate 64 0x7f + b256 = B.replicate 256 0x7f + b1024 = B.replicate 1024 0x7f diff --git a/Crypto/Number/Basic.hs b/Crypto/Number/Basic.hs new file mode 100644 --- /dev/null +++ b/Crypto/Number/Basic.hs @@ -0,0 +1,88 @@ +{-# LANGUAGE BangPatterns #-} +-- | +-- Module : Crypto.Number.Basic +-- License : BSD-style +-- Maintainer : Vincent Hanquez +-- Stability : experimental +-- Portability : Good + +module Crypto.Number.Basic + ( sqrti + , gcde + , gcde_binary + , areEven + ) where + +import Data.Bits + +-- | sqrti returns two integer (l,b) so that l <= sqrt i <= b +-- the implementation is quite naive, use an approximation for the first number +-- and use a dichotomy algorithm to compute the bound relatively efficiently. +sqrti :: Integer -> (Integer, Integer) +sqrti i + | i < 0 = error "cannot compute negative square root" + | i == 0 = (0,0) + | i == 1 = (1,1) + | i == 2 = (1,2) + | otherwise = loop x0 + where + nbdigits = length $ show i + x0n = (if even nbdigits then nbdigits - 2 else nbdigits - 1) `div` 2 + x0 = if even nbdigits then 2 * 10 ^ x0n else 6 * 10 ^ x0n + loop x = case compare (sq x) i of + LT -> iterUp x + EQ -> (x, x) + GT -> iterDown x + iterUp lb = if sq ub >= i then iter lb ub else iterUp ub + where ub = lb * 2 + iterDown ub = if sq lb >= i then iterDown lb else iter lb ub + where lb = ub `div` 2 + iter lb ub + | lb == ub = (lb, ub) + | lb+1 == ub = (lb, ub) + | otherwise = + let d = (ub - lb) `div` 2 in + if sq (lb + d) >= i + then iter lb (ub-d) + else iter (lb+d) ub + sq a = a * a + +-- | get the extended GCD of two integer using integer divMod +gcde :: Integer -> Integer -> (Integer, Integer, Integer) +gcde a b = if d < 0 then (-x,-y,-d) else (x,y,d) where + (d, x, y) = f (a,1,0) (b,0,1) + f t (0, _, _) = t + f (a', sa, ta) t@(b', sb, tb) = + let (q, r) = a' `divMod` b' in + f t (r, sa - (q * sb), ta - (q * tb)) + +-- | get the extended GCD of two integer using the extended binary algorithm (HAC 14.61) +-- get (x,y,d) where d = gcd(a,b) and x,y satisfying ax + by = d +gcde_binary :: Integer -> Integer -> (Integer, Integer, Integer) +gcde_binary a' b' + | b' == 0 = (1,0,a') + | a' >= b' = compute a' b' + | otherwise = (\(x,y,d) -> (y,x,d)) $ compute b' a' + where + getEvenMultiplier !g !x !y + | areEven [x,y] = getEvenMultiplier (g `shiftL` 1) (x `shiftR` 1) (y `shiftR` 1) + | otherwise = (x,y,g) + halfLoop !x !y !u !i !j + | areEven [u,i,j] = halfLoop x y (u `shiftR` 1) (i `shiftR` 1) (j `shiftR` 1) + | even u = halfLoop x y (u `shiftR` 1) ((i + y) `shiftR` 1) ((j - x) `shiftR` 1) + | otherwise = (u, i, j) + compute a b = + let (x,y,g) = getEvenMultiplier 1 a b in + loop g x y x y 1 0 0 1 + + loop g _ _ 0 !v _ _ !c !d = (c, d, g * v) + loop g x y !u !v !a !b !c !d = + let (u2,a2,b2) = halfLoop x y u a b + (v2,c2,d2) = halfLoop x y v c d + in if u2 >= v2 + then loop g x y (u2 - v2) v2 (a2 - c2) (b2 - d2) c2 d2 + else loop g x y u2 (v2 - u2) a2 b2 (c2 - a2) (d2 - b2) + +-- | check if a list of integer are all even +areEven :: [Integer] -> Bool +areEven = and . map even diff --git a/Crypto/Number/Generate.hs b/Crypto/Number/Generate.hs new file mode 100644 --- /dev/null +++ b/Crypto/Number/Generate.hs @@ -0,0 +1,35 @@ +-- | +-- Module : Crypto.Number.Generate +-- License : BSD-style +-- Maintainer : Vincent Hanquez +-- Stability : experimental +-- Portability : Good + +module Crypto.Number.Generate + ( generateMax + , generateBetween + , generateOfSize + ) where + +import Crypto.Number.Serialize +import Crypto.Random.API +import qualified Data.ByteString as B +import Data.Bits ((.|.)) + +-- | generate a positive integer between 0 and m. +-- using as many bytes as necessary to the same size as m, that are converted to integer. +generateMax :: CPRG g => g -> Integer -> (Integer, g) +generateMax rng m = withRandomBytes rng (lengthBytes m) $ \bs -> + os2ip bs `mod` m + +-- | generate a number between the inclusive bound [low,high]. +generateBetween :: CPRG g => g -> Integer -> Integer -> (Integer, g) +generateBetween rng low high = (low + v, rng') + where (v, rng') = generateMax rng (high - low + 1) + +-- | generate a positive integer of a specific size in bits. +-- the number of bits need to be multiple of 8. It will always returns +-- an integer that is close to 2^(1+bits/8) by setting the 2 highest bits to 1. +generateOfSize :: CPRG g => g -> Int -> (Integer, g) +generateOfSize rng bits = withRandomBytes rng (bits `div` 8) $ \bs -> + os2ip $ snd $ B.mapAccumL (\acc w -> (0, w .|. acc)) 0xc0 bs diff --git a/Crypto/Number/ModArithmetic.hs b/Crypto/Number/ModArithmetic.hs new file mode 100644 --- /dev/null +++ b/Crypto/Number/ModArithmetic.hs @@ -0,0 +1,67 @@ +{-# LANGUAGE BangPatterns #-} +{-# LANGUAGE DeriveDataTypeable #-} +-- | +-- Module : Crypto.Number.ModArithmetic +-- License : BSD-style +-- Maintainer : Vincent Hanquez +-- Stability : experimental +-- Portability : Good + +module Crypto.Number.ModArithmetic + ( exponantiation_rtl_binary + , exponantiation + , inverse + , inverseCoprimes + ) where + +import Control.Exception (throw, Exception) +import Crypto.Number.Basic (gcde_binary) +import Data.Bits +import Data.Typeable + +-- | Raised when two numbers are supposed to be coprimes but are not. +data CoprimesAssertionError = CoprimesAssertionError + deriving (Show,Typeable) + +instance Exception CoprimesAssertionError + +-- note on exponantiation: 0^0 is treated as 1 for mimicking the standard library; +-- the mathematic debate is still open on whether or not this is true, but pratically +-- in computer science it shouldn't be useful for anything anyway. + +-- | exponantiation_rtl_binary computes modular exponantiation as b^e mod m +-- using the right-to-left binary exponentiation algorithm (HAC 14.79) +exponantiation_rtl_binary :: Integer -> Integer -> Integer -> Integer +exponantiation_rtl_binary 0 0 m = 1 `mod` m +exponantiation_rtl_binary b e m = loop e b 1 + where sq x = (x * x) `mod` m + loop !0 _ !a = a `mod` m + loop !i !s !a = loop (i `shiftR` 1) (sq s) (if odd i then a * s else a) + +-- | exponantiation computes modular exponantiation as b^e mod m +-- using repetitive squaring. +exponantiation :: Integer -> Integer -> Integer -> Integer +exponantiation b e m + | b == 1 = b + | e == 0 = 1 + | e == 1 = b `mod` m + | even e = let p = (exponantiation b (e `div` 2) m) `mod` m + in (p^(2::Integer)) `mod` m + | otherwise = (b * exponantiation b (e-1) m) `mod` m + +-- | inverse computes the modular inverse as in g^(-1) mod m +inverse :: Integer -> Integer -> Maybe Integer +inverse g m = if d > 1 then Nothing else Just (x `mod` m) + where (x,_,d) = gcde_binary g m + +-- | Compute the modular inverse of 2 coprime numbers. +-- This is equivalent to inverse except that the result +-- is known to exists. +-- +-- if the numbers are not defined as coprime, this function +-- will raise a CoprimesAssertionError. +inverseCoprimes :: Integer -> Integer -> Integer +inverseCoprimes g m = + case inverse g m of + Nothing -> throw CoprimesAssertionError + Just i -> i diff --git a/Crypto/Number/Polynomial.hs b/Crypto/Number/Polynomial.hs new file mode 100644 --- /dev/null +++ b/Crypto/Number/Polynomial.hs @@ -0,0 +1,133 @@ +{-# LANGUAGE BangPatterns #-} +-- | +-- Module : Crypto.Number.Polynomial +-- License : BSD-style +-- Maintainer : Vincent Hanquez +-- Stability : experimental +-- Portability : Good + +module Crypto.Number.Polynomial + ( Monomial(..) + -- * polynomial operations + , Polynomial + , toList + , fromList + , addPoly + , subPoly + , mulPoly + , squarePoly + , expPoly + , divPoly + , negPoly + ) where + +import Data.List (intercalate, sort) +import Data.Vector ((!), Vector) +import qualified Data.Vector as V +import Control.Arrow (first) + +data Monomial = Monomial {-# UNPACK #-} !Int !Integer + deriving (Eq) + +data Polynomial = Polynomial (Vector Monomial) + deriving (Eq) + +instance Ord Monomial where + compare (Monomial w1 v1) (Monomial w2 v2) = + case compare w1 w2 of + EQ -> compare v1 v2 + r -> r + +instance Show Monomial where + show (Monomial w v) = show v ++ "x^" ++ show w + +instance Show Polynomial where + show (Polynomial p) = intercalate "+" $ map show $ V.toList p + +toList :: Polynomial -> [Monomial] +toList (Polynomial p) = V.toList p + +fromList :: [Monomial] -> Polynomial +fromList = Polynomial . V.fromList . reverse . sort . filterZero + where + filterZero = filter (\(Monomial _ v) -> v /= 0) + +getWeight :: Polynomial -> Int -> Maybe Integer +getWeight (Polynomial p) n = look 0 + where + plen = V.length p + look !i + | i >= plen = Nothing + | otherwise = + let (Monomial w v) = p ! i in + case compare w n of + LT -> Nothing + EQ -> Just v + GT -> look (i+1) + + +mergePoly :: (Integer -> Integer -> Integer) -> Polynomial -> Polynomial -> Polynomial +mergePoly f (Polynomial p1) (Polynomial p2) = fromList $ loop 0 0 + where + l1 = V.length p1 + l2 = V.length p2 + loop !i1 !i2 + | i1 == l1 && i2 == l2 = [] + | i1 == l1 = (p2 ! i2) : loop i1 (i2+1) + | i2 == l2 = (p1 ! i1) : loop (i1+1) i2 + | otherwise = + let (coef, i1inc, i2inc) = addCoef (p1 ! i1) (p2 ! i2) in + coef : loop (i1+i1inc) (i2+i2inc) + addCoef m1@(Monomial w1 v1) (Monomial w2 v2) = + case compare w1 w2 of + LT -> (Monomial w2 (f 0 v2), 0, 1) + EQ -> (Monomial w1 (f v1 v2), 1, 1) + GT -> (m1, 1, 0) + +addPoly :: Polynomial -> Polynomial -> Polynomial +addPoly = mergePoly (+) + +subPoly :: Polynomial -> Polynomial -> Polynomial +subPoly = mergePoly (-) + +negPoly :: Polynomial -> Polynomial +negPoly (Polynomial p) = Polynomial $ V.map negateMonomial p + where negateMonomial (Monomial w v) = Monomial w (-v) + +mulPoly :: Polynomial -> Polynomial -> Polynomial +mulPoly p1@(Polynomial v1) p2@(Polynomial v2) = + fromList $ filter (\(Monomial _ v) -> v /= 0) $ map (\i -> Monomial i (c i)) $ reverse [0..(m+n)] + where + (Monomial m _) = v1 ! 0 + (Monomial n _) = v2 ! 0 + c r = foldl (\acc i -> (b $ r-i) * (a $ i) + acc) 0 [0..r] + where + a = maybe 0 id . getWeight p1 + b = maybe 0 id . getWeight p2 + +squarePoly :: Polynomial -> Polynomial +squarePoly p = p `mulPoly` p + +expPoly :: Polynomial -> Integer -> Polynomial +expPoly p e = loop p e + where + loop t 0 = t + loop t n = loop (squarePoly t) (n-1) + +divPoly :: Polynomial -> Polynomial -> (Polynomial, Polynomial) +divPoly p1 p2@(Polynomial pp2) = first fromList $ divLoop p1 + where divLoop d1@(Polynomial pp1) + | V.null pp1 = ([], d1) + | otherwise = + let (Monomial w1 v1) = pp1 ! 0 in + let (Monomial w2 v2) = pp2 ! 0 in + let w = w1 - w2 in + let (v,r) = v1 `divMod` v2 in + if w >= 0 && r == 0 + then + let mono = (Monomial w v) in + let remain = d1 `subPoly` (p2 `mulPoly` (fromList [mono])) in + let (l, finalRem) = divLoop remain in + (mono : l, finalRem) + else + ([], d1) diff --git a/Crypto/Number/Prime.hs b/Crypto/Number/Prime.hs new file mode 100644 --- /dev/null +++ b/Crypto/Number/Prime.hs @@ -0,0 +1,197 @@ +{-# 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 diff --git a/Crypto/Number/Serialize.hs b/Crypto/Number/Serialize.hs new file mode 100644 --- /dev/null +++ b/Crypto/Number/Serialize.hs @@ -0,0 +1,73 @@ +module Crypto.Number.Serialize + ( i2osp + , os2ip + , i2ospOf + , i2ospOf_ + , lengthBytes + ) where + +import Data.ByteString (ByteString) +import qualified Data.ByteString as B +import qualified Data.ByteString.Internal as B +import Data.Bits +import Foreign.Storable +import Foreign.Ptr + +{-# INLINE divMod256 #-} +divMod256 :: Integer -> (Integer, Integer) +divMod256 n = (n `shiftR` 8, n .&. 0xff) + +-- | os2ip converts a byte string into a positive integer +{-# INLINE os2ip #-} +os2ip :: ByteString -> Integer +os2ip = B.foldl' (\a b -> (256 * a) .|. (fromIntegral b)) 0 + +-- | i2osp converts a positive integer into a byte string +i2osp :: Integer -> ByteString +i2osp m + | m < 0 = error "i2osp: cannot convert a negative integer to a bytestring" + | otherwise = B.reverse $ B.unfoldr fdivMod256 m + where fdivMod256 0 = Nothing + fdivMod256 n = Just (fromIntegral a,b) where (b,a) = divMod256 n + + +-- | just like i2osp, but take an extra parameter for size. +-- if the number is too big to fit in @len bytes, nothing is returned +-- otherwise the number is padded with 0 to fit the @len required. +-- +-- FIXME: use unsafeCreate to fill the bytestring +i2ospOf :: Int -> Integer -> Maybe ByteString +i2ospOf len m + | lenbytes < len = Just $ B.replicate (len - lenbytes) 0 `B.append` bytes + | lenbytes == len = Just bytes + | otherwise = Nothing + where + lenbytes = B.length bytes + bytes = i2osp m + +-- | just like i2ospOf except that it doesn't expect a failure. +-- for example if you just took a modulo of the number that represent +-- the size (example the RSA modulo n). +{-# INLINE i2ospOf_ #-} +i2ospOf_ :: Int -> Integer -> ByteString +i2ospOf_ len m = B.unsafeCreate len fillPtr + where fillPtr srcPtr = loop m (srcPtr `plusPtr` (len-1)) + where loop n ptr = do + let (nn,a) = divMod256 n + poke ptr (fromIntegral a) + if ptr == srcPtr + then return () + else (if nn == 0 then fillerLoop else loop nn) (ptr `plusPtr` (-1)) + fillerLoop ptr = do + poke ptr 0 + if ptr == srcPtr + then return () + else fillerLoop (ptr `plusPtr` (-1)) + +-- | returns the number of bytes to store an integer with i2osp +-- +-- FIXME: really slow implementation. use log or bigger shifts. +lengthBytes :: Integer -> Int +lengthBytes n + | n < 256 = 1 + | otherwise = 1 + lengthBytes (n `shiftR` 8) diff --git a/LICENSE b/LICENSE new file mode 100644 --- /dev/null +++ b/LICENSE @@ -0,0 +1,24 @@ +Copyright (c) 2010-2012 Vincent Hanquez + +All rights reserved. + +Redistribution and use in source and binary forms, with or without +modification, are permitted provided that the following conditions +are met: +1. Redistributions of source code must retain the above copyright + notice, this list of conditions and the following disclaimer. +2. Redistributions in binary form must reproduce the above copyright + notice, this list of conditions and the following disclaimer in the + documentation and/or other materials provided with the distribution. + +THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND +ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE +IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE +ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHORS OR CONTRIBUTORS BE LIABLE +FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL +DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS +OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) +HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT +LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY +OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF +SUCH DAMAGE. diff --git a/Setup.hs b/Setup.hs new file mode 100644 --- /dev/null +++ b/Setup.hs @@ -0,0 +1,2 @@ +import Distribution.Simple +main = defaultMain diff --git a/Tests/RNG.hs b/Tests/RNG.hs new file mode 100644 --- /dev/null +++ b/Tests/RNG.hs @@ -0,0 +1,38 @@ +module RNG where + +import Data.Word +import Data.List (foldl') +import qualified Data.ByteString as B +import Crypto.Random.API +import Control.Arrow (first) + +{- this is a just test rng. this is absolutely not a serious RNG. DO NOT use elsewhere -} +data Rng = Rng (Int, Int) + +getByte :: Rng -> (Word8, Rng) +getByte (Rng (mz, mw)) = (r, g) + where mz2 = 36969 * (mz `mod` 65536) + mw2 = 18070 * (mw `mod` 65536) + r = fromIntegral (mz2 + mw2) + g = Rng (mz2, mw2) + +getBytes :: Int -> Rng -> ([Word8], Rng) +getBytes 0 g = ([], g) +getBytes n g = + let (b, g') = getByte g + (l, g'') = getBytes (n-1) g' + in (b:l, g'') + +instance CPRG Rng where + cprgGenBytes g len = first B.pack $ getBytes len g + cprgSupplyEntropy g e = reseed e g + cprgNeedReseed _ = maxBound + +reseed :: B.ByteString -> Rng -> Rng +reseed bs (Rng (a,b)) = Rng (fromIntegral a', b') + where a' = foldl' (\v i -> ((fromIntegral v) + (fromIntegral i) * 36969) `mod` 65536) a l + b' = foldl' (\v i -> ((fromIntegral v) + (fromIntegral i) * 18070) `mod` 65536) b l + l = B.unpack bs + +rng :: Rng +rng = Rng (1,2) diff --git a/Tests/Tests.hs b/Tests/Tests.hs new file mode 100644 --- /dev/null +++ b/Tests/Tests.hs @@ -0,0 +1,135 @@ +{-# LANGUAGE OverloadedStrings #-} +{-# LANGUAGE ViewPatterns #-} + +import Test.Framework (defaultMain, testGroup) +import Test.Framework.Providers.QuickCheck2 (testProperty) +import Test.Framework.Providers.HUnit (testCase) + +import Test.QuickCheck +import Test.HUnit +--import Test.QuickCheck.Test + +import Control.Applicative ((<$>)) + +import qualified Data.ByteString as B + +import Crypto.Number.ModArithmetic +import Crypto.Number.Basic +import Crypto.Number.Generate +import Crypto.Number.Prime +import Crypto.Number.Serialize + +import RNG + +prop_gcde_binary_valid :: (Positive Integer, Positive Integer) -> Bool +prop_gcde_binary_valid (Positive a, Positive b) = + and [v==v', a*x' + b*y' == v', a*x + b*y == v, gcd a b == v] + where (x,y,v) = gcde_binary a b + (x',y',v') = gcde a b + +prop_modexp_rtl_valid :: (NonNegative Integer, + NonNegative Integer, + Positive Integer) + -> Bool +prop_modexp_rtl_valid (NonNegative a, NonNegative b, Positive m) = + exponantiation_rtl_binary a b m == ((a ^ b) `mod` m) + +prop_modinv_valid :: (Positive Integer, Positive Integer) -> Bool +prop_modinv_valid (Positive a, Positive m) + | m > 1 = case inverse a m of + Just ainv -> (ainv * a) `mod` m == 1 + Nothing -> True + | otherwise = True + +prop_sqrti_valid :: Positive Integer -> Bool +prop_sqrti_valid (Positive i) = l*l <= i && i <= u*u where (l, u) = sqrti i + +prop_generate_prime_valid :: Seed -> Bool +prop_generate_prime_valid i = + -- because of the next naive test, we can't generate easily number above 32 bits + -- otherwise it slows down the tests to uselessness. when AKS or ECPP is implemented + -- we can revisit the number here + primalityTestNaive $ withRNG i (\g -> generatePrime g 32) + +prop_miller_rabin_valid :: (Seed, PositiveSmall) -> Bool +prop_miller_rabin_valid (seed, PositiveSmall i) + | i <= 3 = True + | otherwise = + -- miller rabin only returns with certitude that the integer is composite. + let b = withRNG seed (\g -> isProbablyPrime g i) + in (b == False && primalityTestNaive i == False) || b == True + +prop_generate_valid :: (Seed, Positive Integer) -> Bool +prop_generate_valid (seed, Positive h) = + let v = withRNG seed (\g -> generateMax g h) + in (v >= 0 && v < h) + +withAleasInteger :: Rng -> Seed -> (Rng -> (a,Rng)) -> a +withAleasInteger g (Seed i) f = fst $ f $ reseed (i2osp $ fromIntegral i) g + +withRNG :: Seed -> (Rng -> (a,Rng)) -> a +withRNG seed f = withAleasInteger rng seed f + +newtype PositiveSmall = PositiveSmall Integer + deriving (Show,Eq) + +instance Arbitrary PositiveSmall where + arbitrary = PositiveSmall . fromIntegral <$> (resize (2^(20 :: Int)) (arbitrary :: Gen Int)) + +data Range = Range Integer Integer + deriving (Show,Eq) + +instance Arbitrary Range where + arbitrary = do (Positive x) <- arbitrary :: Gen (Positive Int) + (Positive r) <- arbitrary :: Gen (Positive Int) + return $ Range (fromIntegral x) (fromIntegral r) + +newtype Seed = Seed Integer + deriving (Eq) + +instance Show Seed where + show s = "Seed " ++ show s + +instance Arbitrary Seed where + arbitrary = arbitrary >>= \(Positive i) -> return (Seed i) + +serializationKATTests = concatMap f vectors + where f (v, bs) = [ testCase ("i2osp " ++ show v) (i2osp v @=? bs) + , testCase ("os2ip " ++ show v) (os2ip bs @=? v) + ] + vectors = + [ (0x10000, "\SOH\NUL\NUL") + , (0x1234, "\DC24") + , (0xf123456, "\SI\DC24V") + , (0xf21908421feabd21490, "\SI!\144\132!\254\171\210\DC4\144") + , (0x7521908421feabd21490, "u!\144\132!\254\171\210\DC4\144") + ] + +main :: IO () +main = defaultMain + [ testGroup "serialization" + [ testProperty "unbinary.binary==id" (\(Positive i) -> os2ip (i2osp i) == i) + , testProperty "length integer" (\(Positive i) -> B.length (i2osp i) == lengthBytes i) + , testGroup "KAT" serializationKATTests + ] + , testGroup "gcde binary" + [ testProperty "gcde" prop_gcde_binary_valid + ] + , testGroup "exponantiation" + [ testProperty "right-to-left" prop_modexp_rtl_valid + ] + , testGroup "inverse" + [ testProperty "inverse" prop_modinv_valid + ] + , testGroup "sqrt integer" + [ testProperty "sqrt" prop_sqrti_valid + ] + , testGroup "generation" + [ testProperty "max" prop_generate_valid + --, testProperty "between" (\seed (Range l h) -> let generated = withRNG seed (\rng -> generateBetween rng l (l+h)) + -- in (generated > l && generated < h)) + ] + , testGroup "primality test" + [ testProperty "miller-rabin" prop_miller_rabin_valid + ] + ] diff --git a/crypto-numbers.cabal b/crypto-numbers.cabal new file mode 100644 --- /dev/null +++ b/crypto-numbers.cabal @@ -0,0 +1,58 @@ +Name: crypto-numbers +Version: 0.1.0 +Description: Cryptographic numbers: functions and algorithms +License: BSD3 +License-file: LICENSE +Copyright: Vincent Hanquez +Author: Vincent Hanquez +Maintainer: Vincent Hanquez +Synopsis: Cryptographic numbers: functions and algorithms +Category: Cryptography +Build-Type: Simple +Homepage: http://github.com/vincenthz/hs-crypto-numbers +Cabal-Version: >=1.8 +Extra-Source-Files: Tests/*.hs + +Library + Build-Depends: base >= 4 && < 5 + , bytestring + , vector + , crypto-random-api + Exposed-modules: Crypto.Number.ModArithmetic + Crypto.Number.Serialize + Crypto.Number.Generate + Crypto.Number.Basic + Crypto.Number.Polynomial + Crypto.Number.Prime + ghc-options: -Wall + +Test-Suite test-crypto-numbers + type: exitcode-stdio-1.0 + hs-source-dirs: Tests + Main-Is: Tests.hs + Build-depends: base >= 4 && < 5 + , crypto-random-api + , crypto-numbers + , bytestring + , vector + , QuickCheck >= 2 + , HUnit + , test-framework >= 0.3.3 + , test-framework-quickcheck2 >= 0.2.9 + , test-framework-hunit + ghc-options: -Wall -O2 + +Benchmark bench-crypto-numbers + hs-source-dirs: Benchmarks + Main-Is: Benchmarks.hs + type: exitcode-stdio-1.0 + Build-depends: base >= 4 && < 5 + , bytestring + , crypto-random-api + , crypto-numbers + , criterion + , mtl + +source-repository head + type: git + location: git://github.com/vincenthz/hs-crypto-numbers