Galois Field

An efficient implementation of Galois fields used in cryptography research.

Technical background

A Galois field GF(p^q), for prime p and positive q, is a field (GF(p^q), +, *, 0, 1) of finite order. Explicitly,

• (GF(p^q), +, 0) is an abelian group,
• (GF(p^q) \ {0}, *, 1) is an abelian group,
• * is distributive over +, and
• #GF(p^q) is finite.

Prime fields

Any Galois field has a unique characteristic p, the minimum positive p such that p(1) = 1 + … + 1 = 0, and p is prime. The smallest Galois field of characteristic p is a prime field, and any Galois field of characteristic p is a finite-dimensional vector space over its prime subfield.

For example, GF(4) is a Galois field of characteristic 2 that is a two-dimensional vector space over the prime subfield GF(2) = Z / 2Z.

Extension fields

Any Galois field has order a prime power p^q for prime p and positive q, and there is a Galois field GF(p^q) of any prime power order p^q that is unique up to non-unique isomorphism. Any Galois field GF(p^q) can be constructed as an extension field over a smaller Galois subfield GF(p^r), through the identification GF(p^q) = GF(p^r)[X] / <f(X)> for an irreducible monic splitting polynomial f(X) of degree q - r + 1 in the polynomial ring GF(p^r)[X].

For example, GF(4) has order 2^2 and can be constructed as an extension field GF(2)[X] / <f(X)> where f(X) = X^2 + X + 1 is an irreducible monic splitting quadratic polynomial in GF(2)[X].

Binary fields

A Galois field of the form GF(2^m) for big positive m is a sum of x^n for a non-empty set of 0 < n < m. For computational efficiency in cryptography, an element of a binary field can be represented by an integer that represents a bit string.

For example, X^8 + X^4 + X^3 + X + 1 can be represented as the integer 283 that represents the bit string 100011011.

Example usage

Include the following required language extensions.

{-# LANGUAGE DataKinds #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE MultiParamTypeClasses #-}

Import the following functions at minimum.

import PrimeField (PrimeField)
import ExtensionField (ExtensionField, IrreducibleMonic(split), fromList, t, x)

Prime fields

The following type declaration creates a prime field of a given characteristic.

type Fq = PrimeField 21888242871839275222246405745257275088696311157297823662689037894645226208583

Note that the characteristic given must be prime.

Galois field arithmetic can then be performed in this prime field.

fq :: Fq
fq = 5216004179354450092383934373463611881445186046129513844852096383579774061693

fq' :: Fq
fq' = 10757805228921058098980668000791497318123219899766237205512608761387909753942

arithmeticFq :: (Fq, Fq, Fq, Fq)
arithmeticFq = (fq + fq', fq - fq', fq * fq', fq / fq')

Extension fields

The following data type declaration creates a splitting polynomial given an irreducible monic polynomial.

data P2
instance IrreducibleMonic Fq P2 where
  split _ = x^2 + 1

The following type declaration then creates an extension field with this splitting polynomial.

type Fq2 = ExtensionField Fq P2

Note that the splitting polynomial given must be irreducible and monic in the prime field.

Similarly, further extension fields can be constructed iteratively as follows.

data P6
instance IrreducibleMonic Fq2 P6 where
  split _ = x^3 - (9 + t x)

type Fq6 = ExtensionField Fq2 P6

data P12
instance IrreducibleMonic Fq6 P12 where
  split _ = x^2 - t x

type Fq12 = ExtensionField Fq6 P12

Note that x accesses the current indeterminate variable and t descends the tower of indeterminate variables.

Galois field arithmetic can then be performed in this extension field.

fq12 :: Fq12
fq12 = fromList
  [ fromList
    [ fromList
      [ 4025484419428246835913352650763180341703148406593523188761836807196412398582
      , 5087667423921547416057913184603782240965080921431854177822601074227980319916
      ]
    , fromList
      [ 8868355606921194740459469119392835913522089996670570126495590065213716724895
      , 12102922015173003259571598121107256676524158824223867520503152166796819430680
      ]
    , fromList
      [ 92336131326695228787620679552727214674825150151172467042221065081506740785
      , 5482141053831906120660063289735740072497978400199436576451083698548025220729
      ]
    ]
  , fromList
    [ fromList
      [ 7642691434343136168639899684817459509291669149586986497725240920715691142493
      , 1211355239100959901694672926661748059183573115580181831221700974591509515378
      ]
    , fromList
      [ 20725578899076721876257429467489710434807801418821512117896292558010284413176
      , 17642016461759614884877567642064231230128683506116557502360384546280794322728
      ]
    , fromList
      [ 17449282511578147452934743657918270744212677919657988500433959352763226500950
      , 1205855382909824928004884982625565310515751070464736233368671939944606335817
      ]
    ]
  ]

fq12' :: Fq12
fq12' = fromList
  [ fromList
    [ fromList
      [ 495492586688946756331205475947141303903957329539236899715542920513774223311
      , 9283314577619389303419433707421707208215462819919253486023883680690371740600
      ]
    , fromList
      [ 11142072730721162663710262820927009044232748085260948776285443777221023820448
      , 1275691922864139043351956162286567343365697673070760209966772441869205291758
      ]
    , fromList
      [ 20007029371545157738471875537558122753684185825574273033359718514421878893242
      , 9839139739201376418106411333971304469387172772449235880774992683057627654905
      ]
    ]
  , fromList
    [ fromList
      [ 9503058454919356208294350412959497499007919434690988218543143506584310390240
      , 19236630380322614936323642336645412102299542253751028194541390082750834966816
      ]
    , fromList
      [ 18019769232924676175188431592335242333439728011993142930089933693043738917983
      , 11549213142100201239212924317641009159759841794532519457441596987622070613872
      ]
    , fromList
      [ 9656683724785441232932664175488314398614795173462019188529258009817332577664
      , 20666848762667934776817320505559846916719041700736383328805334359135638079015
      ]
    ]
  ]

arithmeticFq12 :: (Fq12, Fq12, Fq12, Fq12)
arithmeticFq12 = (fq12 + fq12', fq12 - fq12', fq12 * fq12', fq12 / fq12')

Note that

a + bx + (c + dx)y + (e + fx)y^2 + (g + hx + (i + jx)y + (k + lx)y^2)z

where x, y, z is a tower of indeterminate variables is constructed by

fromList [ fromList [fromList [a, b], fromList [c, d], fromList [e, f]]
         , fromList [fromList [g, h], fromList [i, j], fromList [k, l]] ] :: Fq12

Binary fields

The following type declaration creates a binary field modulo a given splitting irreducible binary polynomial.

type F2m = BinaryField 0x80000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000425

Note that the splitting polynomial given must be irreducible in F2.

Galois field arithmetic can then be performed in this binary field.

f2m :: F2m
f2m = 0x303001d34b856296c16c0d40d3cd7750a93d1d2955fa80aa5f40fc8db7b2abdbde53950f4c0d293cdd711a35b67fb1499ae60038614f1394abfa3b4c850d927e1e7769c8eec2d19

f2m' :: F2m
f2m' = 0x37bf27342da639b6dccfffeb73d69d78c6c27a6009cbbca1980f8533921e8a684423e43bab08a576291af8f461bb2a8b3531d2f0485c19b16e2f1516e23dd3c1a4827af1b8ac15b

arithmeticF2m :: (F2m, F2m, F2m, F2m)
arithmeticF2m = (f2m + f2m', f2m - f2m', f2m * f2m', f2m / f2m')

License

Copyright (c) 2019 Adjoint Inc.

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