crypton-1.1.4: Crypto/Number/ModArithmetic.hs
{-# LANGUAGE BangPatterns #-}
-- |
-- Module : Crypto.Number.ModArithmetic
-- License : BSD-style
-- Maintainer : Vincent Hanquez <vincent@snarc.org>
-- Stability : experimental
-- Portability : Good
module Crypto.Number.ModArithmetic (
-- * Exponentiation
expSafe,
expFast,
-- * Inverse computing
inverse,
inverseCoprimes,
inverseFermat,
-- * Squares
jacobi,
squareRoot,
) where
import Control.Exception (Exception, throw)
import Crypto.Number.Basic
import Crypto.Number.Compat
-- | Raised when two numbers are supposed to be coprimes but are not.
data CoprimesAssertionError = CoprimesAssertionError
deriving (Show)
instance Exception CoprimesAssertionError
-- | Compute the modular exponentiation of base^exponent using
-- algorithms design to avoid side channels and timing measurement
--
-- Modulo need to be odd otherwise the normal fast modular exponentiation
-- is used.
--
-- When used with integer-simple, this function is not different
-- from expFast, and thus provide the same unstudied and dubious
-- timing and side channels claims.
--
-- Before GHC 8.4.2, powModSecInteger is missing from integer-gmp,
-- so expSafe has the same security as expFast.
expSafe
:: Integer
-- ^ base
-> Integer
-- ^ exponent
-> Integer
-- ^ modulo
-> Integer
-- ^ result
expSafe b e m
| odd m =
gmpPowModSecInteger b e m
`onGmpUnsupported` ( gmpPowModInteger b e m
`onGmpUnsupported` exponentiation b e m
)
| otherwise =
gmpPowModInteger b e m
`onGmpUnsupported` exponentiation b e m
-- | Compute the modular exponentiation of base^exponent using
-- the fastest algorithm without any consideration for
-- hiding parameters.
--
-- Use this function when all the parameters are public,
-- otherwise 'expSafe' should be preferred.
expFast
:: Integer
-- ^ base
-> Integer
-- ^ exponent
-> Integer
-- ^ modulo
-> Integer
-- ^ result
expFast b e m = gmpPowModInteger b e m `onGmpUnsupported` exponentiation b e m
-- | @exponentiation@ computes modular exponentiation as /b^e mod m/
-- using repetitive squaring.
exponentiation :: Integer -> Integer -> Integer -> Integer
exponentiation b e m
| b == 1 = b
| e == 0 = 1
| e == 1 = b `mod` m
| even e =
let p = exponentiation b (e `div` 2) m `mod` m
in (p ^ (2 :: Integer)) `mod` m
| otherwise = (b * exponentiation 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 = gmpInverse g m `onGmpUnsupported` v
where
v
| d > 1 = Nothing
| otherwise = Just (x `mod` m)
(x, _, d) = gcde g m
-- | Compute the modular inverse of two 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
-- | Computes the Jacobi symbol (a/n).
-- 0 ≤ a < n; n ≥ 3 and odd.
--
-- The Legendre and Jacobi symbols are indistinguishable exactly when the
-- lower argument is an odd prime, in which case they have the same value.
--
-- See algorithm 2.149 in "Handbook of Applied Cryptography" by Alfred J. Menezes et al.
jacobi :: Integer -> Integer -> Maybe Integer
jacobi a n
| n < 3 || even n = Nothing
| a == 0 || a == 1 = Just a
| n <= a = jacobi (a `mod` n) n
| a < 0 =
let b = if n `mod` 4 == 1 then 1 else -1
in fmap (* b) (jacobi (-a) n)
| otherwise =
let (e, a1) = asPowerOf2AndOdd a
nMod8 = n `mod` 8
nMod4 = n `mod` 4
a1Mod4 = a1 `mod` 4
s' = if even e || nMod8 == 1 || nMod8 == 7 then 1 else -1
s = if nMod4 == 3 && a1Mod4 == 3 then -s' else s'
n1 = n `mod` a1
in if a1 == 1
then Just s
else fmap (* s) (jacobi n1 a1)
-- | Modular inverse using Fermat's little theorem. This works only when
-- the modulus is prime but avoids side channels like in 'expSafe'.
inverseFermat :: Integer -> Integer -> Integer
inverseFermat g p = expSafe g (p - 2) p
-- | Raised when the assumption about the modulus is invalid.
data ModulusAssertionError = ModulusAssertionError
deriving (Show)
instance Exception ModulusAssertionError
-- | Modular square root of @g@ modulo a prime @p@.
--
-- If the modulus is found not to be prime, the function will raise a
-- 'ModulusAssertionError'.
--
-- This implementation is variable time and should be used with public
-- parameters only.
squareRoot :: Integer -> Integer -> Maybe Integer
squareRoot p
| p < 2 = throw ModulusAssertionError
| otherwise =
case p `divMod` 8 of
(v, 3) -> method1 (2 * v + 1)
(v, 7) -> method1 (2 * v + 2)
(u, 5) -> method2 u
(_, 1) -> tonelliShanks p
(0, 2) -> \a -> Just (if even a then 0 else 1)
_ -> throw ModulusAssertionError
where
x `eqMod` y = (x - y) `mod` p == 0
validate g y
| (y * y) `eqMod` g = Just y
| otherwise = Nothing
-- p == 4u + 3 and u' == u + 1
method1 u' g =
let y = expFast g u' p
in validate g y
-- p == 8u + 5
method2 u g =
let gamma = expFast (2 * g) u p
g_gamma = g * gamma
i = (2 * g_gamma * gamma) `mod` p
y = (g_gamma * (i - 1)) `mod` p
in validate g y
tonelliShanks :: Integer -> Integer -> Maybe Integer
tonelliShanks p a
| aa == 0 = Just 0
| otherwise =
case expFast aa p2 p of
b
| b == p1 -> Nothing
| b == 1 ->
Just $
go
(expFast aa ((s + 1) `div` 2) p)
(expFast aa s p)
(expFast n s p)
e
| otherwise -> throw ModulusAssertionError
where
aa = a `mod` p
p1 = p - 1
p2 = p1 `div` 2
n = findN 2
x `mul` y = (x * y) `mod` p
pow2m 0 x = x
pow2m i x = pow2m (i - 1) (x `mul` x)
(e, s) = asPowerOf2AndOdd p1
-- find a quadratic non-residue
findN i
| expFast i p2 p == p1 = i
| otherwise = findN (i + 1)
-- find m such that b^(2^m) == 1 (mod p)
findM b i
| b == 1 = i
| otherwise = findM (b `mul` b) (i + 1)
go !x b g !r
| b == 1 = x
| otherwise =
let r' = findM b 0
z = pow2m (r - r' - 1) g
x' = x `mul` z
b' = b `mul` g'
g' = z `mul` z
in go x' b' g' r'