Fix tests and provide documentation for Crypto.Number.F2m
This commit is contained in:
Bodigrim
parent
e80eaa56f3
commit
66ae77e805
@ -9,6 +9,7 @@
|
||||
-- not optimal and it doesn't provide protection against timing
|
||||
-- attacks. The 'm' parameter is implicitly derived from the irreducible
|
||||
-- polynomial where applicable.
|
||||
|
||||
module Crypto.Number.F2m
|
||||
( BinaryPolynomial
|
||||
, addF2m
|
||||
@ -22,88 +23,125 @@ module Crypto.Number.F2m
|
||||
|
||||
import Data.Bits ((.&.),(.|.),xor,shift,testBit)
|
||||
import Crypto.Number.Basic
|
||||
import Crypto.Internal.Imports
|
||||
|
||||
-- | Binary Polynomial represented by an integer
|
||||
type BinaryPolynomial = Integer
|
||||
|
||||
-- | Addition over F₂m. This is just a synonym of 'xor'.
|
||||
addF2m :: Integer -> Integer -> Integer
|
||||
-- | Addition over F₂m. This is just a synonym of 'xor'.
|
||||
addF2m :: Integer
|
||||
-> Integer
|
||||
-> Integer
|
||||
addF2m = xor
|
||||
{-# INLINE addF2m #-}
|
||||
|
||||
-- | Binary polynomial reduction modulo using long division algorithm.
|
||||
modF2m :: BinaryPolynomial -- ^ Irreducible binary polynomial
|
||||
-> Integer -> Integer
|
||||
modF2m fx = go
|
||||
where
|
||||
lfx = log2 fx
|
||||
go n | s == 0 = n `xor` fx
|
||||
| s < 0 = n
|
||||
| otherwise = go $ n `xor` shift fx s
|
||||
-- | Reduction by modulo over F₂m.
|
||||
--
|
||||
-- This function is undefined for negative arguments, because their bit
|
||||
-- representation is platform-dependent. Zero modulus is also prohibited.
|
||||
modF2m :: BinaryPolynomial -- ^ Modulus
|
||||
-> Integer
|
||||
-> Integer
|
||||
modF2m fx i
|
||||
| fx < 0 || i < 0 = error "modF2m: negative number represent no binary polynomial"
|
||||
| fx == 0 = error "modF2m: cannot divide by zero polynomial"
|
||||
| fx == 1 = 0
|
||||
| otherwise = go i
|
||||
where
|
||||
s = log2 n - lfx
|
||||
lfx = log2 fx
|
||||
go n | s == 0 = n `addF2m` fx
|
||||
| s < 0 = n
|
||||
| otherwise = go $ n `addF2m` shift fx s
|
||||
where s = log2 n - lfx
|
||||
{-# INLINE modF2m #-}
|
||||
|
||||
-- | Multiplication over F₂m.
|
||||
--
|
||||
-- n1 * n2 (in F(2^m))
|
||||
mulF2m :: BinaryPolynomial -- ^ Irreducible binary polynomial
|
||||
-> Integer -> Integer -> Integer
|
||||
mulF2m fx n1 n2 = modF2m fx
|
||||
$ go (if n2 `mod` 2 == 1 then n1 else 0) (log2 n2)
|
||||
where
|
||||
go n s | s == 0 = n
|
||||
| otherwise = if testBit n2 s
|
||||
then go (n `xor` shift n1 s) (s - 1)
|
||||
else go n (s - 1)
|
||||
-- This function is undefined for negative arguments, because their bit
|
||||
-- representation is platform-dependent. Zero modulus is also prohibited.
|
||||
mulF2m :: BinaryPolynomial -- ^ Modulus
|
||||
-> Integer
|
||||
-> Integer
|
||||
-> Integer
|
||||
mulF2m fx n1 n2
|
||||
| fx < 0
|
||||
|| n1 < 0
|
||||
|| n2 < 0 = error "mulF2m: negative number represent no binary binary polynomial"
|
||||
| fx == 0 = error "modF2m: cannot multiply modulo zero polynomial"
|
||||
| otherwise = modF2m fx $ go (if n2 `mod` 2 == 1 then n1 else 0) (log2 n2)
|
||||
where
|
||||
go n s | s == 0 = n
|
||||
| otherwise = if testBit n2 s
|
||||
then go (n `addF2m` shift n1 s) (s - 1)
|
||||
else go n (s - 1)
|
||||
{-# INLINABLE mulF2m #-}
|
||||
|
||||
-- | Squaring over F₂m.
|
||||
--
|
||||
-- This function is undefined for negative arguments, because their bit
|
||||
-- representation is platform-dependent. Zero modulus is also prohibited.
|
||||
--
|
||||
-- TODO: This is still slower than @mulF2m@.
|
||||
|
||||
--
|
||||
-- Multiplication table? C?
|
||||
squareF2m :: BinaryPolynomial -- ^ Irreducible binary polynomial
|
||||
-> Integer -> Integer
|
||||
squareF2m :: BinaryPolynomial -- ^ Modulus
|
||||
-> Integer
|
||||
-> Integer
|
||||
squareF2m fx = modF2m fx . squareF2m'
|
||||
{-# INLINE squareF2m #-}
|
||||
|
||||
squareF2m' :: Integer -> Integer
|
||||
squareF2m' n1 = go n1 ln1
|
||||
where
|
||||
ln1 = log2 n1
|
||||
go n s | s == 0 = n
|
||||
| otherwise = go (x .|. y) (s - 1)
|
||||
-- | Squaring over F₂m.
|
||||
--
|
||||
-- This function is undefined for negative arguments, because their bit
|
||||
-- representation is platform-dependent.
|
||||
squareF2m' :: Integer
|
||||
-> Integer
|
||||
squareF2m' n1
|
||||
| n1 < 0 = error "mulF2m: negative number represent no binary binary polynomial"
|
||||
| otherwise = go n1 ln1
|
||||
where
|
||||
x = shift (shift n (2 * (s - ln1) - 1)) (2 * (ln1 - s) + 2)
|
||||
y = n .&. (shift 1 (2 * (ln1 - s) + 1) - 1)
|
||||
ln1 = log2 n1
|
||||
go n s | s == 0 = n
|
||||
| otherwise = go (x .|. y) (s - 1)
|
||||
where
|
||||
x = shift (shift n (2 * (s - ln1) - 1)) (2 * (ln1 - s) + 2)
|
||||
y = n .&. (shift 1 (2 * (ln1 - s) + 1) - 1)
|
||||
{-# INLINE squareF2m' #-}
|
||||
|
||||
-- | Inversion of @n over F₂m using extended Euclidean algorithm.
|
||||
-- | Extended GCD algorithm for polynomials. For @a@ and @b@ returns @(g, u, v)@ such that @a * u + b * v == g@.
|
||||
--
|
||||
-- If @n doesn't have an inverse, Nothing is returned.
|
||||
invF2m :: BinaryPolynomial -- ^ Irreducible binary polynomial
|
||||
-> Integer -> Maybe Integer
|
||||
invF2m _ 0 = Nothing
|
||||
invF2m fx n
|
||||
| n >= fx = Nothing
|
||||
| otherwise = go n fx 1 0
|
||||
where
|
||||
go u v g1 g2
|
||||
| u == 1 = Just $ modF2m fx g1
|
||||
| j < 0 = go u (v `xor` shift u (-j)) g1 (g2 `xor` shift g1 (-j))
|
||||
| otherwise = go (u `xor` shift v j) v (g1 `xor` shift g2 j) g2
|
||||
where
|
||||
j = log2 u - log2 v
|
||||
-- Reference: https://en.wikipedia.org/wiki/Polynomial_greatest_common_divisor#B.C3.A9zout.27s_identity_and_extended_GCD_algorithm
|
||||
gcdF2m :: Integer
|
||||
-> Integer
|
||||
-> (Integer, Integer, Integer)
|
||||
gcdF2m a b = go (a, b, 1, 0, 0, 1)
|
||||
where
|
||||
go (g, 0, u, _, v, _)
|
||||
= (g, u, v)
|
||||
go (r0, r1, s0, s1, t0, t1)
|
||||
= go (r1, r0 `addF2m` shift r1 j, s1, s0 `addF2m` shift s1 j, t1, t0 `addF2m` shift t1 j)
|
||||
where j = max 0 (log2 r0 - log2 r1)
|
||||
|
||||
-- | Modular inversion over F₂m.
|
||||
-- If @n@ doesn't have an inverse, 'Nothing' is returned.
|
||||
--
|
||||
-- This function is undefined for negative arguments, because their bit
|
||||
-- representation is platform-dependent. Zero modulus is also prohibited.
|
||||
invF2m :: BinaryPolynomial -- ^ Modulus
|
||||
-> Integer
|
||||
-> Maybe Integer
|
||||
invF2m fx n = if g == 1 then Just (modF2m fx u) else Nothing
|
||||
where
|
||||
(g, u, _) = gcdF2m n fx
|
||||
{-# INLINABLE invF2m #-}
|
||||
|
||||
-- | Division over F₂m. If the dividend doesn't have an inverse it returns
|
||||
-- 'Nothing'.
|
||||
--
|
||||
-- Compute n1 / n2
|
||||
divF2m :: BinaryPolynomial -- ^ Irreducible binary polynomial
|
||||
-> Integer -- ^ Dividend
|
||||
-> Integer -- ^ Quotient
|
||||
-> Maybe Integer
|
||||
-- This function is undefined for negative arguments, because their bit
|
||||
-- representation is platform-dependent. Zero modulus is also prohibited.
|
||||
divF2m :: BinaryPolynomial -- ^ Modulus
|
||||
-> Integer -- ^ Dividend
|
||||
-> Integer -- ^ Divisor
|
||||
-> Maybe Integer -- ^ Quotient
|
||||
divF2m fx n1 n2 = mulF2m fx n1 <$> invF2m fx n2
|
||||
{-# INLINE divF2m #-}
|
||||
|
||||
Loading…
Reference in New Issue
Block a user