129 lines
4.3 KiB
Haskell
129 lines
4.3 KiB
Haskell
{-# LANGUAGE BangPatterns #-}
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{-# LANGUAGE DeriveDataTypeable #-}
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-- |
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-- Module : Crypto.Number.ModArithmetic
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-- License : BSD-style
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-- Maintainer : Vincent Hanquez <vincent@snarc.org>
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-- Stability : experimental
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-- Portability : Good
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module Crypto.Number.ModArithmetic
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(
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-- * Exponentiation
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expSafe
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, expFast
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-- * Inverse computing
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, inverse
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, inverseCoprimes
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, jacobi
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, inverseFermat
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) where
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import Control.Exception (throw, Exception)
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import Crypto.Number.Basic
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import Crypto.Number.Compat
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-- | Raised when two numbers are supposed to be coprimes but are not.
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data CoprimesAssertionError = CoprimesAssertionError
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deriving (Show)
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instance Exception CoprimesAssertionError
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-- | Compute the modular exponentiation of base^exponent using
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-- algorithms design to avoid side channels and timing measurement
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--
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-- Modulo need to be odd otherwise the normal fast modular exponentiation
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-- is used.
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--
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-- When used with integer-simple, this function is not different
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-- from expFast, and thus provide the same unstudied and dubious
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-- timing and side channels claims.
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--
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-- Before GHC 8.4.2, powModSecInteger is missing from integer-gmp,
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-- so expSafe has the same security as expFast.
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expSafe :: Integer -- ^ base
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-> Integer -- ^ exponent
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-> Integer -- ^ modulo
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-> Integer -- ^ result
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expSafe b e m
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| odd m = gmpPowModSecInteger b e m `onGmpUnsupported`
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(gmpPowModInteger b e m `onGmpUnsupported`
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exponentiation b e m)
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| otherwise = gmpPowModInteger b e m `onGmpUnsupported`
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exponentiation b e m
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-- | Compute the modular exponentiation of base^exponent using
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-- the fastest algorithm without any consideration for
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-- hiding parameters.
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--
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-- Use this function when all the parameters are public,
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-- otherwise 'expSafe' should be preferred.
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expFast :: Integer -- ^ base
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-> Integer -- ^ exponent
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-> Integer -- ^ modulo
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-> Integer -- ^ result
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expFast b e m = gmpPowModInteger b e m `onGmpUnsupported` exponentiation b e m
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-- | @exponentiation@ computes modular exponentiation as /b^e mod m/
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-- using repetitive squaring.
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exponentiation :: Integer -> Integer -> Integer -> Integer
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exponentiation b e m
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| b == 1 = b
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| e == 0 = 1
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| e == 1 = b `mod` m
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| even e = let p = (exponentiation b (e `div` 2) m) `mod` m
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in (p^(2::Integer)) `mod` m
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| otherwise = (b * exponentiation b (e-1) m) `mod` m
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-- | @inverse@ computes the modular inverse as in /g^(-1) mod m/.
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inverse :: Integer -> Integer -> Maybe Integer
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inverse g m = gmpInverse g m `onGmpUnsupported` v
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where
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v
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| d > 1 = Nothing
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| otherwise = Just (x `mod` m)
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(x,_,d) = gcde g m
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-- | Compute the modular inverse of two coprime numbers.
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-- This is equivalent to inverse except that the result
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-- is known to exists.
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--
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-- If the numbers are not defined as coprime, this function
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-- will raise a 'CoprimesAssertionError'.
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inverseCoprimes :: Integer -> Integer -> Integer
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inverseCoprimes g m =
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case inverse g m of
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Nothing -> throw CoprimesAssertionError
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Just i -> i
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-- | Computes the Jacobi symbol (a/n).
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-- 0 ≤ a < n; n ≥ 3 and odd.
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--
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-- The Legendre and Jacobi symbols are indistinguishable exactly when the
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-- lower argument is an odd prime, in which case they have the same value.
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--
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-- See algorithm 2.149 in "Handbook of Applied Cryptography" by Alfred J. Menezes et al.
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jacobi :: Integer -> Integer -> Maybe Integer
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jacobi a n
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| n < 3 || even n = Nothing
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| a == 0 || a == 1 = Just a
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| n <= a = jacobi (a `mod` n) n
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| a < 0 =
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let b = if n `mod` 4 == 1 then 1 else -1
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in fmap (*b) (jacobi (-a) n)
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| otherwise =
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let (e, a1) = asPowerOf2AndOdd a
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nMod8 = n `mod` 8
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nMod4 = n `mod` 4
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a1Mod4 = a1 `mod` 4
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s' = if even e || nMod8 == 1 || nMod8 == 7 then 1 else -1
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s = if nMod4 == 3 && a1Mod4 == 3 then -s' else s'
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n1 = n `mod` a1
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in if a1 == 1 then Just s
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else fmap (*s) (jacobi n1 a1)
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-- | Modular inverse using Fermat's little theorem. This works only when
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-- the modulus is prime but avoids side channels like in 'expSafe'.
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inverseFermat :: Integer -> Integer -> Integer
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inverseFermat g p = expSafe g (p - 2) p
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