Crypto.Number.F2m

crypton-0.33: Cryptography Primitives sink

License | BSD-style
Maintainer | Danny Navarro <j@dannynavarro.net>
Stability | experimental
Portability | Good
Safe Haskell | Safe-Inferred
Language | Haskell2010

Crypto.Number.F2m

Description

This module provides basic arithmetic operations over F₂m. Performance is not optimal and it doesn't provide protection against timing attacks. The `m` parameter is implicitly derived from the irreducible polynomial where applicable.

Synopsis

- type BinaryPolynomial = Integer
- addF2m :: Integer -> Integer -> Integer
- mulF2m :: BinaryPolynomial -> Integer -> Integer -> Integer
- squareF2m' :: Integer -> Integer
- squareF2m :: BinaryPolynomial -> Integer -> Integer
- powF2m :: BinaryPolynomial -> Integer -> Integer -> Integer
- modF2m :: BinaryPolynomial -> Integer -> Integer
- sqrtF2m :: BinaryPolynomial -> Integer -> Integer
- invF2m :: BinaryPolynomial -> Integer -> Maybe Integer
- divF2m :: BinaryPolynomial -> Integer -> Integer -> Maybe Integer

# Documentation

type BinaryPolynomial = Integer #

Binary Polynomial represented by an integer

addF2m :: Integer -> Integer -> Integer #

Addition over F₂m. This is just a synonym of `xor`.

mulF2m #

Arguments

:: BinaryPolynomial | Modulus
-> Integer |
-> Integer |
-> Integer |

Multiplication over F₂m.

This function is undefined for negative arguments, because their bit representation is platform-dependent. Zero modulus is also prohibited.

squareF2m' :: Integer -> Integer #

Squaring over F₂m without reduction by modulo.

The implementation utilizes the fact that for binary polynomial S(x) we have S(x)^2 = S(x^2). In other words, insert a zero bit between every bits of argument: 1101 -> 1010001.

This function is undefined for negative arguments, because their bit representation is platform-dependent.

squareF2m #

Arguments

:: BinaryPolynomial | Modulus
-> Integer |
-> Integer |

Squaring over F₂m.

This function is undefined for negative arguments, because their bit representation is platform-dependent. Zero modulus is also prohibited.

powF2m #

Arguments

:: BinaryPolynomial | Modulus
-> Integer | a
-> Integer | b
-> Integer |

Exponentiation in F₂m by computing `a^b mod fx`.

This implements an exponentiation by squaring based solution. It inherits the same restrictions as `squareF2m`. Negative exponents are disallowed.

modF2m #

Arguments

:: BinaryPolynomial | Modulus
-> Integer |
-> Integer |

Reduction by modulo over F₂m.

This function is undefined for negative arguments, because their bit representation is platform-dependent. Zero modulus is also prohibited.

sqrtF2m #

Arguments

:: BinaryPolynomial | Modulus
-> Integer | a
-> Integer |

Square rooot in F₂m.

We exploit the fact that `a^(2^m) = a`, or in particular, `a^(2^m - 1) = 1` from a classical result by Lagrange. Thus the square root is simply `a^(2^(m - 1))`.

invF2m #

Arguments

:: BinaryPolynomial | Modulus
-> Integer |
-> Maybe Integer |

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.

divF2m #

Arguments

:: BinaryPolynomial | Modulus
-> Integer | Dividend
-> Integer | Divisor
-> Maybe Integer | Quotient

Division over F₂m. If the dividend doesn't have an inverse it returns `Nothing`.

This function is undefined for negative arguments, because their bit representation is platform-dependent. Zero modulus is also prohibited.