diff --git a/ChangeLog.md b/ChangeLog.md
--- a/ChangeLog.md
+++ b/ChangeLog.md
@@ -1,5 +1,13 @@
# Change log for galois-field
+## 1.0.2
+
+* Add `Mod` dependency for prime fields.
+* Add minor optimisations for prime fields.
+* Remove unsafeCoerce for number coercions.
+* Bump poly dependency to 0.4.
+* Bump upper bound for protolude.
+
## 1.0.1
* Add `Bit` dependency for binary fields.
diff --git a/README.md b/README.md
deleted file mode 100644
--- a/README.md
+++ /dev/null
@@ -1,211 +0,0 @@
-
-
- 
-
-
-
-[](https://circleci.com/gh/adjoint-io/galois-field)
-[](https://hackage.haskell.org/package/galois-field)
-
-# 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] / \ for an *irreducible monic 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] / \ where f(X) = X^2 + X + 1 is an irreducible monic 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. It should always be used when the field characteristic is 2.
-
-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.
-```haskell
-{-# LANGUAGE DataKinds #-}
-{-# LANGUAGE FlexibleInstances #-}
-{-# LANGUAGE MultiParamTypeClasses #-}
-{-# LANGUAGE OverloadedLists #-}
-{-# LANGUAGE PatternSynonyms #-}
-```
-Import the following functions at minimum.
-```haskell
-import Data.Field.Galois (Prime, Extension, IrreducibleMonic(poly), Binary,
- pattern X, pattern X2, pattern X3, pattern Y)
-```
-
-### Prime fields
-
-The following type declaration creates a prime field of a given characteristic.
-```haskell
-type Fq = Prime 21888242871839275222246405745257275088696311157297823662689037894645226208583
-```
-Note that the characteristic given *must* be prime.
-
-Galois field arithmetic can then be performed in this prime field.
-```haskell
-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 polynomial given an irreducible monic polynomial.
-```haskell
-data P2
-instance IrreducibleMonic P2 Fq where
- poly _ = X2 + 1
-```
-The following type declaration then creates an extension field with this polynomial.
-```haskell
-type Fq2 = Extension P2 Fq
-```
-Note that the polynomial given *must* be irreducible and monic in the prime field.
-
-Similarly, further extension fields can be constructed iteratively as follows.
-```haskell
-data P6
-instance IrreducibleMonic P6 Fq2 where
- poly _ = X3 - (9 + Y X)
-
-type Fq6 = Extension P6 Fq2
-
-data P12
-instance IrreducibleMonic P12 Fq6 where
- poly _ = X2 - Y X
-
-type Fq12 = Extension P12 Fq6
-```
-Note that `X, X2, X3` accesses the current indeterminate variables and `Y` descends the tower of indeterminate variables.
-
-Galois field arithmetic can then be performed in this extension field.
-```haskell
-fq12 :: Fq12
-fq12 =
- [ [ [ 4025484419428246835913352650763180341703148406593523188761836807196412398582
- , 5087667423921547416057913184603782240965080921431854177822601074227980319916
- ]
- , [ 8868355606921194740459469119392835913522089996670570126495590065213716724895
- , 12102922015173003259571598121107256676524158824223867520503152166796819430680
- ]
- , [ 92336131326695228787620679552727214674825150151172467042221065081506740785
- , 5482141053831906120660063289735740072497978400199436576451083698548025220729
- ]
- ]
- , [ [ 7642691434343136168639899684817459509291669149586986497725240920715691142493
- , 1211355239100959901694672926661748059183573115580181831221700974591509515378
- ]
- , [ 20725578899076721876257429467489710434807801418821512117896292558010284413176
- , 17642016461759614884877567642064231230128683506116557502360384546280794322728
- ]
- , [ 17449282511578147452934743657918270744212677919657988500433959352763226500950
- , 1205855382909824928004884982625565310515751070464736233368671939944606335817
- ]
- ]
- ]
-
-fq12' :: Fq12
-fq12' =
- [ [ [ 495492586688946756331205475947141303903957329539236899715542920513774223311
- , 9283314577619389303419433707421707208215462819919253486023883680690371740600
- ]
- , [ 11142072730721162663710262820927009044232748085260948776285443777221023820448
- , 1275691922864139043351956162286567343365697673070760209966772441869205291758
- ]
- , [ 20007029371545157738471875537558122753684185825574273033359718514421878893242
- , 9839139739201376418106411333971304469387172772449235880774992683057627654905
- ]
- ]
- , [ [ 9503058454919356208294350412959497499007919434690988218543143506584310390240
- , 19236630380322614936323642336645412102299542253751028194541390082750834966816
- ]
- , [ 18019769232924676175188431592335242333439728011993142930089933693043738917983
- , 11549213142100201239212924317641009159759841794532519457441596987622070613872
- ]
- , [ 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
-```haskell
-[ [ [a, b], [c, d], [e, f] ]
-, [ [g, h], [i, j], [k, l] ] ] :: Fq12
-```
-
-### Binary fields
-
-The following type declaration creates a binary field modulo a given irreducible binary polynomial.
-```haskell
-type F2m = Binary 0x80000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000425
-```
-Note that the polynomial given *must* be irreducible in F2.
-
-Galois field arithmetic can then be performed in this binary field.
-```haskell
-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.
-
-Permission is hereby granted, free of charge, to any person obtaining a copy
-of this software and associated documentation files (the "Software"), to deal
-in the Software without restriction, including without limitation the rights
-to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
-copies of the Software, and to permit persons to whom the Software is
-furnished to do so, subject to the following conditions:
-
-The above copyright notice and this permission notice shall be included in all
-copies or substantial portions of the Software.
-
-THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
-EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
-MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.
-IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM,
-DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR
-OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE
-OR OTHER DEALINGS IN THE SOFTWARE.
-```
diff --git a/README.notex.md b/README.notex.md
new file mode 100644
--- /dev/null
+++ b/README.notex.md
@@ -0,0 +1,211 @@
+
+
+
+
+
+
+[](https://circleci.com/gh/adjoint-io/galois-field)
+[](https://hackage.haskell.org/package/galois-field)
+
+# 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] / \ for an *irreducible monic 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] / \ where f(X) = X^2 + X + 1 is an irreducible monic 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. It should always be used when the field characteristic is 2.
+
+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.
+```haskell
+{-# LANGUAGE DataKinds #-}
+{-# LANGUAGE FlexibleInstances #-}
+{-# LANGUAGE MultiParamTypeClasses #-}
+{-# LANGUAGE OverloadedLists #-}
+{-# LANGUAGE PatternSynonyms #-}
+```
+Import the following functions at minimum.
+```haskell
+import Data.Field.Galois (Prime, Extension, IrreducibleMonic(poly), Binary,
+ pattern X, pattern X2, pattern X3, pattern Y)
+```
+
+### Prime fields
+
+The following type declaration creates a prime field of a given characteristic.
+```haskell
+type Fq = Prime 21888242871839275222246405745257275088696311157297823662689037894645226208583
+```
+Note that the characteristic given *must* be prime.
+
+Galois field arithmetic can then be performed in this prime field.
+```haskell
+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 polynomial given an irreducible monic polynomial.
+```haskell
+data P2
+instance IrreducibleMonic P2 Fq where
+ poly _ = X2 + 1
+```
+The following type declaration then creates an extension field with this polynomial.
+```haskell
+type Fq2 = Extension P2 Fq
+```
+Note that the polynomial given *must* be irreducible and monic in the prime field.
+
+Similarly, further extension fields can be constructed iteratively as follows.
+```haskell
+data P6
+instance IrreducibleMonic P6 Fq2 where
+ poly _ = X3 - (9 + Y X)
+
+type Fq6 = Extension P6 Fq2
+
+data P12
+instance IrreducibleMonic P12 Fq6 where
+ poly _ = X2 - Y X
+
+type Fq12 = Extension P12 Fq6
+```
+Note that `X, X2, X3` accesses the current indeterminate variables and `Y` descends the tower of indeterminate variables.
+
+Galois field arithmetic can then be performed in this extension field.
+```haskell
+fq12 :: Fq12
+fq12 =
+ [ [ [ 4025484419428246835913352650763180341703148406593523188761836807196412398582
+ , 5087667423921547416057913184603782240965080921431854177822601074227980319916
+ ]
+ , [ 8868355606921194740459469119392835913522089996670570126495590065213716724895
+ , 12102922015173003259571598121107256676524158824223867520503152166796819430680
+ ]
+ , [ 92336131326695228787620679552727214674825150151172467042221065081506740785
+ , 5482141053831906120660063289735740072497978400199436576451083698548025220729
+ ]
+ ]
+ , [ [ 7642691434343136168639899684817459509291669149586986497725240920715691142493
+ , 1211355239100959901694672926661748059183573115580181831221700974591509515378
+ ]
+ , [ 20725578899076721876257429467489710434807801418821512117896292558010284413176
+ , 17642016461759614884877567642064231230128683506116557502360384546280794322728
+ ]
+ , [ 17449282511578147452934743657918270744212677919657988500433959352763226500950
+ , 1205855382909824928004884982625565310515751070464736233368671939944606335817
+ ]
+ ]
+ ]
+
+fq12' :: Fq12
+fq12' =
+ [ [ [ 495492586688946756331205475947141303903957329539236899715542920513774223311
+ , 9283314577619389303419433707421707208215462819919253486023883680690371740600
+ ]
+ , [ 11142072730721162663710262820927009044232748085260948776285443777221023820448
+ , 1275691922864139043351956162286567343365697673070760209966772441869205291758
+ ]
+ , [ 20007029371545157738471875537558122753684185825574273033359718514421878893242
+ , 9839139739201376418106411333971304469387172772449235880774992683057627654905
+ ]
+ ]
+ , [ [ 9503058454919356208294350412959497499007919434690988218543143506584310390240
+ , 19236630380322614936323642336645412102299542253751028194541390082750834966816
+ ]
+ , [ 18019769232924676175188431592335242333439728011993142930089933693043738917983
+ , 11549213142100201239212924317641009159759841794532519457441596987622070613872
+ ]
+ , [ 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
+```haskell
+[ [ [a, b], [c, d], [e, f] ]
+, [ [g, h], [i, j], [k, l] ] ] :: Fq12
+```
+
+### Binary fields
+
+The following type declaration creates a binary field modulo a given irreducible binary polynomial.
+```haskell
+type F2m = Binary 0x80000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000425
+```
+Note that the polynomial given *must* be irreducible in F2.
+
+Galois field arithmetic can then be performed in this binary field.
+```haskell
+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) 2019-2020 Adjoint Inc.
+
+Permission is hereby granted, free of charge, to any person obtaining a copy
+of this software and associated documentation files (the "Software"), to deal
+in the Software without restriction, including without limitation the rights
+to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
+copies of the Software, and to permit persons to whom the Software is
+furnished to do so, subject to the following conditions:
+
+The above copyright notice and this permission notice shall be included in all
+copies or substantial portions of the Software.
+
+THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
+EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
+MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.
+IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM,
+DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR
+OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE
+OR OTHER DEALINGS IN THE SOFTWARE.
+```
diff --git a/galois-field.cabal b/galois-field.cabal
--- a/galois-field.cabal
+++ b/galois-field.cabal
@@ -1,114 +1,176 @@
-cabal-version: 1.12
+cabal-version: 1.12
-- This file has been generated from package.yaml by hpack version 0.31.2.
--
-- see: https://github.com/sol/hpack
--
--- hash: f4e2a6cce087c45e10262060054544b7596de8e4a173935c13f460d1bf83ddd3
+-- hash: 4f1f517ac1812c3cb4f6bd4c78c3091393d6c7708ce11feb2729ac6f871f92c4
-name: galois-field
-version: 1.0.1
-synopsis: Galois field library
-description: An efficient implementation of Galois fields used in cryptography research
-category: Cryptography
-homepage: https://github.com/adjoint-io/galois-field#readme
-bug-reports: https://github.com/adjoint-io/galois-field/issues
-maintainer: Adjoint Inc (info@adjoint.io)
-license: MIT
-license-file: LICENSE
-build-type: Simple
+name: galois-field
+version: 1.0.2
+synopsis: Galois field library
+description:
+ An efficient implementation of Galois fields used in cryptography research
+
+category: Cryptography
+homepage: https://github.com/adjoint-io/galois-field#readme
+bug-reports: https://github.com/adjoint-io/galois-field/issues
+maintainer: Adjoint Inc (info@adjoint.io)
+license: MIT
+license-file: LICENSE
+build-type: Simple
extra-source-files:
- README.md
- ChangeLog.md
+ README.notex.md
+ ChangeLog.md
source-repository head
- type: git
+ type: git
location: https://github.com/adjoint-io/galois-field
library
- exposed-modules:
- Data.Field.Galois
+ exposed-modules: Data.Field.Galois
other-modules:
- Data.Field.Galois.Base
- Data.Field.Galois.Binary
- Data.Field.Galois.Extension
- Data.Field.Galois.Frobenius
- Data.Field.Galois.Prime
- Data.Field.Galois.Sqrt
- Data.Field.Galois.Tower
- Data.Field.Galois.Unity
- hs-source-dirs:
- src
- default-extensions: LambdaCase RecordWildCards OverloadedStrings NoImplicitPrelude FlexibleInstances FlexibleContexts ScopedTypeVariables RankNTypes DataKinds DeriveFunctor DeriveGeneric GeneralizedNewtypeDeriving KindSignatures MultiParamTypeClasses OverloadedLists PatternSynonyms TypeFamilies
- ghc-options: -freverse-errors -O2 -Wall
+ Data.Field.Galois.Base
+ Data.Field.Galois.Binary
+ Data.Field.Galois.Extension
+ Data.Field.Galois.Frobenius
+ Data.Field.Galois.Prime
+ Data.Field.Galois.Sqrt
+ Data.Field.Galois.Tower
+ Data.Field.Galois.Unity
+
+ hs-source-dirs: src
+ default-extensions:
+ NoImplicitPrelude
+ DataKinds
+ DeriveFunctor
+ DeriveGeneric
+ FlexibleContexts
+ FlexibleInstances
+ GeneralizedNewtypeDeriving
+ KindSignatures
+ LambdaCase
+ MultiParamTypeClasses
+ OverloadedLists
+ OverloadedStrings
+ PatternSynonyms
+ RankNTypes
+ RecordWildCards
+ ScopedTypeVariables
+ TypeFamilies
+
+ ghc-options: -freverse-errors -O2 -Wall
build-depends:
- MonadRandom
- , base >=4.10 && <5
- , bitvec >=1.0.2
- , groups
- , integer-gmp
- , poly >=0.3.2
- , protolude >=0.2 && <0.3
- , semirings >=0.5
- , tasty-quickcheck
- , vector
- , wl-pprint-text
- default-language: Haskell2010
+ base >=4.10 && <5
+ , bitvec >=1.0.2 && <1.1
+ , groups >=0.4.1 && <0.5
+ , integer-gmp >=1.0.2 && <1.1
+ , mod >=0.1.0 && <0.2
+ , MonadRandom >=0.5.1 && <0.6
+ , poly >=0.3.2 && <0.5
+ , protolude >=0.2 && <0.4
+ , QuickCheck >=2.13 && <2.14
+ , semirings >=0.5 && <0.6
+ , vector >=0.12.0 && <0.13
+ , wl-pprint-text >=1.2.0 && <1.3
+ default-language: Haskell2010
+
test-suite galois-field-tests
- type: exitcode-stdio-1.0
- main-is: Main.hs
+ type: exitcode-stdio-1.0
+ main-is: Main.hs
other-modules:
- Test.Binary
- Test.Extension
- Test.Galois
- Test.Prime
- Paths_galois_field
- hs-source-dirs:
- test
- default-extensions: LambdaCase RecordWildCards OverloadedStrings NoImplicitPrelude FlexibleInstances FlexibleContexts ScopedTypeVariables RankNTypes DataKinds DeriveFunctor DeriveGeneric GeneralizedNewtypeDeriving KindSignatures MultiParamTypeClasses OverloadedLists PatternSynonyms TypeFamilies
- ghc-options: -freverse-errors -O2 -Wall -main-is Main
+ Paths_galois_field
+ Test.Binary
+ Test.Extension
+ Test.Galois
+ Test.Prime
+
+ hs-source-dirs: test
+ default-extensions:
+ NoImplicitPrelude
+ DataKinds
+ DeriveFunctor
+ DeriveGeneric
+ FlexibleContexts
+ FlexibleInstances
+ GeneralizedNewtypeDeriving
+ KindSignatures
+ LambdaCase
+ MultiParamTypeClasses
+ OverloadedLists
+ OverloadedStrings
+ PatternSynonyms
+ RankNTypes
+ RecordWildCards
+ ScopedTypeVariables
+ TypeFamilies
+
+ ghc-options: -freverse-errors -O2 -Wall -main-is Main
build-depends:
- MonadRandom
- , base >=4.10 && <5
- , bitvec >=1.0.2
+ base >=4.10 && <5
+ , bitvec >=1.0.2 && <1.1
, galois-field
- , groups
- , integer-gmp
- , poly >=0.3.2
- , protolude >=0.2 && <0.3
- , semirings >=0.5
- , tasty
- , tasty-quickcheck
- , vector
- , wl-pprint-text
- default-language: Haskell2010
+ , groups >=0.4.1 && <0.5
+ , integer-gmp >=1.0.2 && <1.1
+ , mod >=0.1.0 && <0.2
+ , MonadRandom >=0.5.1 && <0.6
+ , poly >=0.3.2 && <0.5
+ , protolude >=0.2 && <0.4
+ , QuickCheck >=2.13 && <2.14
+ , semirings >=0.5 && <0.6
+ , tasty >=1.2 && <1.3
+ , tasty-quickcheck >=0.10 && <0.11
+ , vector >=0.12.0 && <0.13
+ , wl-pprint-text >=1.2.0 && <1.3
+ default-language: Haskell2010
+
benchmark galois-field-benchmarks
- type: exitcode-stdio-1.0
- main-is: Main.hs
+ type: exitcode-stdio-1.0
+ main-is: Main.hs
other-modules:
- Bench.Binary
- Bench.Extension
- Bench.Galois
- Bench.Prime
- Paths_galois_field
- hs-source-dirs:
- bench
- default-extensions: LambdaCase RecordWildCards OverloadedStrings NoImplicitPrelude FlexibleInstances FlexibleContexts ScopedTypeVariables RankNTypes DataKinds DeriveFunctor DeriveGeneric GeneralizedNewtypeDeriving KindSignatures MultiParamTypeClasses OverloadedLists PatternSynonyms TypeFamilies
- ghc-options: -freverse-errors -O2 -Wall -main-is Main
+ Bench.Binary
+ Bench.Extension
+ Bench.Galois
+ Bench.Prime
+ Paths_galois_field
+
+ hs-source-dirs: bench
+ default-extensions:
+ NoImplicitPrelude
+ DataKinds
+ DeriveFunctor
+ DeriveGeneric
+ FlexibleContexts
+ FlexibleInstances
+ GeneralizedNewtypeDeriving
+ KindSignatures
+ LambdaCase
+ MultiParamTypeClasses
+ OverloadedLists
+ OverloadedStrings
+ PatternSynonyms
+ RankNTypes
+ RecordWildCards
+ ScopedTypeVariables
+ TypeFamilies
+
+ ghc-options: -freverse-errors -O2 -Wall -main-is Main
build-depends:
- MonadRandom
- , base >=4.10 && <5
- , bitvec >=1.0.2
- , criterion
+ base >=4.10 && <5
+ , bitvec >=1.0.2 && <1.1
+ , criterion >=1.5 && <1.6
, galois-field
- , groups
- , integer-gmp
- , poly >=0.3.2
- , protolude >=0.2 && <0.3
- , semirings >=0.5
- , tasty-quickcheck
- , vector
- , wl-pprint-text
- default-language: Haskell2010
+ , groups >=0.4.1 && <0.5
+ , integer-gmp >=1.0.2 && <1.1
+ , mod >=0.1.0 && <0.2
+ , MonadRandom >=0.5.1 && <0.6
+ , poly >=0.3.2 && <0.5
+ , protolude >=0.2 && <0.4
+ , QuickCheck >=2.13 && <2.14
+ , semirings >=0.5 && <0.6
+ , vector >=0.12.0 && <0.13
+ , wl-pprint-text >=1.2.0 && <1.3
+
+ default-language: Haskell2010
diff --git a/src/Data/Field/Galois.hs b/src/Data/Field/Galois.hs
--- a/src/Data/Field/Galois.hs
+++ b/src/Data/Field/Galois.hs
@@ -1,22 +1,29 @@
+-- |
+-- An efficient implementation of Galois fields used in cryptography research.
module Data.Field.Galois
- ( module Data.Field
- -- * Galois fields
- , module Data.Field.Galois.Base
- -- ** Prime fields
- , module Data.Field.Galois.Prime
- -- ** Extension fields
- , module Data.Field.Galois.Extension
- -- ** Binary fields
- , module Data.Field.Galois.Binary
- -- ** Square roots
- , module Data.Field.Galois.Sqrt
- -- ** Towers of fields
- , module Data.Field.Galois.Tower
- -- ** Roots of unity
- , module Data.Field.Galois.Unity
- ) where
+ ( -- * Galois fields
+ module Data.Field.Galois.Base,
-import Data.Field
+ -- ** Prime fields
+ module Data.Field.Galois.Prime,
+
+ -- ** Extension fields
+ module Data.Field.Galois.Extension,
+
+ -- ** Binary fields
+ module Data.Field.Galois.Binary,
+
+ -- ** Square roots
+ module Data.Field.Galois.Sqrt,
+
+ -- ** Towers of fields
+ module Data.Field.Galois.Tower,
+
+ -- ** Roots of unity
+ module Data.Field.Galois.Unity,
+ )
+where
+
import Data.Field.Galois.Base
import Data.Field.Galois.Binary
import Data.Field.Galois.Extension
diff --git a/src/Data/Field/Galois/Base.hs b/src/Data/Field/Galois/Base.hs
--- a/src/Data/Field/Galois/Base.hs
+++ b/src/Data/Field/Galois/Base.hs
@@ -8,7 +8,7 @@
import Data.Field (Field)
import qualified Data.Group as G (Group(..))
import GHC.Natural (Natural)
-import Test.Tasty.QuickCheck (Arbitrary)
+import Test.QuickCheck (Arbitrary)
import Text.PrettyPrint.Leijen.Text (Pretty)
-------------------------------------------------------------------------------
diff --git a/src/Data/Field/Galois/Binary.hs b/src/Data/Field/Galois/Binary.hs
--- a/src/Data/Field/Galois/Binary.hs
+++ b/src/Data/Field/Galois/Binary.hs
@@ -18,7 +18,7 @@
import GHC.Exts (IsList(..))
import GHC.Natural (Natural)
import GHC.TypeNats (natVal)
-import Test.Tasty.QuickCheck (Arbitrary(..), choose)
+import Test.QuickCheck (Arbitrary(..), choose)
import Text.PrettyPrint.Leijen.Text (Pretty(..))
import Data.Field.Galois.Base (GaloisField(..))
diff --git a/src/Data/Field/Galois/Extension.hs b/src/Data/Field/Galois/Extension.hs
--- a/src/Data/Field/Galois/Extension.hs
+++ b/src/Data/Field/Galois/Extension.hs
@@ -19,13 +19,13 @@
import Protolude as P hiding (Semiring, rem, toList)
import Control.Monad.Random (Random(..))
-import Data.Euclidean (Euclidean(..), GcdDomain)
+import Data.Euclidean (Euclidean(..), GcdDomain, gcdExt)
import Data.Field (Field)
import Data.Group (Group(..))
-import Data.Poly (VPoly, gcdExt, monomial, toPoly, unPoly)
+import Data.Poly (VPoly, monomial, toPoly, unPoly, scale, leading)
import Data.Semiring (Ring(..), Semiring(..))
import GHC.Exts (IsList(..))
-import Test.Tasty.QuickCheck (Arbitrary(..), vector)
+import Test.QuickCheck (Arbitrary(..), vector)
import Text.PrettyPrint.Leijen.Text (Pretty(..))
import Data.Field.Galois.Base (GaloisField(..))
@@ -103,9 +103,11 @@
-- Extension fields are fractional.
instance IrreducibleMonic p k => Fractional (Extension p k) where
- recip (E x) = case gcdExt x $ poly (witness :: Extension p k) of
- (1, y) -> E y
- _ -> divZeroError
+ recip (E x) = case leading g of
+ Just (0, c) -> E $ scale 0 (recip c) y
+ _ -> divZeroError
+ where
+ (g, y) = gcdExt x $ poly (witness :: Extension p k)
{-# INLINABLE recip #-}
fromRational (x:%y) = fromInteger x / fromInteger y
{-# INLINABLE fromRational #-}
diff --git a/src/Data/Field/Galois/Prime.hs b/src/Data/Field/Galois/Prime.hs
--- a/src/Data/Field/Galois/Prime.hs
+++ b/src/Data/Field/Galois/Prime.hs
@@ -3,7 +3,6 @@
, PrimeField
, fromP
, toP
- , toP'
) where
import Protolude as P hiding (Semiring, natVal, rem)
@@ -12,11 +11,11 @@
import Data.Euclidean as S (Euclidean(..), GcdDomain)
import Data.Field (Field)
import Data.Group (Group(..))
+import Data.Mod (Mod, unMod, (^%))
import Data.Semiring (Ring(..), Semiring(..))
-import GHC.Integer.GMP.Internals (recipModInteger)
-import GHC.Natural (Natural, naturalFromInteger, naturalToInteger, powModNatural)
+import GHC.Natural (naturalToInteger)
import GHC.TypeNats (natVal)
-import Test.Tasty.QuickCheck (Arbitrary(..), choose)
+import Test.QuickCheck (Arbitrary(..), choose)
import Text.PrettyPrint.Leijen.Text (Pretty(..))
import Data.Field.Galois.Base (GaloisField(..))
@@ -32,12 +31,15 @@
fromP :: k -> Integer
-- | Prime field elements.
-newtype Prime (p :: Nat) = P Natural
- deriving (Bits, Eq, Generic, Hashable, NFData, Ord, Show)
+newtype Prime (p :: Nat) = P (Mod p)
+ deriving (Eq, Ord, Show, Generic, Num, Fractional, Euclidean, Field, GcdDomain, Ring, Semiring, Bounded, Enum, NFData)
+instance Hashable (Prime p) where
+ hashWithSalt s (P x) = hashWithSalt s (unMod x)
+
-- Prime fields are convertible.
instance KnownNat p => PrimeField (Prime p) where
- fromP (P x) = naturalToInteger x
+ fromP (P x) = naturalToInteger (unMod x)
{-# INLINABLE fromP #-}
-- Prime fields are Galois fields.
@@ -61,9 +63,7 @@
instance KnownNat p => Group (Prime p) where
invert = recip
{-# INLINE invert #-}
- pow y@(P x) n
- | n >= 0 = P $ powModNatural x (fromIntegral n) $ natVal (witness :: Prime p)
- | otherwise = pow (recip y) $ P.negate n
+ pow (P x) k = P (x ^% k)
{-# INLINE pow #-}
-- Prime fields are multiplicative monoids.
@@ -79,94 +79,14 @@
{-# INLINE stimes #-}
-------------------------------------------------------------------------------
--- Numeric instances
--------------------------------------------------------------------------------
-
--- Prime fields are fractional.
-instance KnownNat p => Fractional (Prime p) where
- recip (P 0) = divZeroError
- recip (P x) = P $ recipModNatural x $ natVal (witness :: Prime p)
- {-# INLINE recip #-}
- fromRational (x:%y) = fromInteger x / fromInteger y
- {-# INLINABLE fromRational #-}
-
--- Prime fields are numeric.
-instance KnownNat p => Num (Prime p) where
- P x + P y = P $ if xy >= p then xy - p else xy
- where
- xy = x + y
- p = natVal (witness :: Prime p)
- {-# INLINE (+) #-}
- P x * P y = P $ rem (x * y) $ natVal (witness :: Prime p)
- {-# INLINE (*) #-}
- P x - P y = P $ if x >= y then x - y else natVal (witness :: Prime p) + x - y
- {-# INLINE (-) #-}
- negate (P 0) = P 0
- negate (P x) = P $ natVal (witness :: Prime p) - x
- {-# INLINE negate #-}
- fromInteger x = P $ naturalFromInteger $ mod x $ naturalToInteger $ natVal (witness :: Prime p)
- {-# INLINABLE fromInteger #-}
- abs = panic "Prime.abs: not implemented."
- signum = panic "Prime.signum: not implemented."
-
--------------------------------------------------------------------------------
--- Semiring instances
--------------------------------------------------------------------------------
-
--- Prime fields are Euclidean domains.
-instance KnownNat p => Euclidean (Prime p) where
- degree = panic "Prime.degree: not implemented."
- quotRem = (flip (,) 0 .) . (/)
- {-# INLINE quotRem #-}
-
--- Prime fields are fields.
-instance KnownNat p => Field (Prime p)
-
--- Prime fields are GCD domains.
-instance KnownNat p => GcdDomain (Prime p)
-
--- Prime fields are rings.
-instance KnownNat p => Ring (Prime p) where
- negate = P.negate
- {-# INLINE negate #-}
-
--- Prime fields are semirings.
-instance KnownNat p => Semiring (Prime p) where
- fromNatural = fromIntegral
- {-# INLINABLE fromNatural #-}
- one = P 1
- {-# INLINE one #-}
- plus = (+)
- {-# INLINE plus #-}
- times = (*)
- {-# INLINE times #-}
- zero = P 0
- {-# INLINE zero #-}
-
--------------------------------------------------------------------------------
-- Other instances
-------------------------------------------------------------------------------
-- Prime fields are arbitrary.
instance KnownNat p => Arbitrary (Prime p) where
- arbitrary = P . naturalFromInteger <$>
- choose (0, naturalToInteger $ natVal (witness :: Prime p) - 1)
+ arbitrary = choose (minBound, maxBound)
{-# INLINABLE arbitrary #-}
--- Prime fields are bounded.
-instance KnownNat p => Bounded (Prime p) where
- maxBound = P $ natVal (witness :: Prime p) - 1
- {-# INLINE maxBound #-}
- minBound = P 0
- {-# INLINE minBound #-}
-
--- Prime fields are enumerable.
-instance KnownNat p => Enum (Prime p) where
- fromEnum = fromIntegral
- {-# INLINABLE fromEnum #-}
- toEnum = fromIntegral
- {-# INLINABLE toEnum #-}
-
-- Prime fields are integral.
instance KnownNat p => Integral (Prime p) where
quotRem = S.quotRem
@@ -176,13 +96,13 @@
-- Prime fields are pretty.
instance KnownNat p => Pretty (Prime p) where
- pretty (P x) = pretty $ naturalToInteger x
+ pretty (P x) = pretty $ naturalToInteger $ unMod x
-- Prime fields are random.
instance KnownNat p => Random (Prime p) where
- random = randomR (P 0, P $ natVal (witness :: Prime p) - 1)
+ random = randomR (minBound, maxBound)
{-# INLINABLE random #-}
- randomR (a, b) = first (P . naturalFromInteger) . randomR (fromP a, fromP b)
+ randomR (a, b) = first fromInteger . randomR (fromP a, fromP b)
{-# INLINABLE randomR #-}
-- Prime fields are real.
@@ -198,18 +118,3 @@
toP :: KnownNat p => Integer -> Prime p
toP = fromInteger
{-# INLINABLE toP #-}
-
--- | Unsafe convert from @Z@ to @GF(p)@.
-toP' :: KnownNat p => Integer -> Prime p
-toP' = P . naturalFromInteger
-{-# INLINABLE toP' #-}
-
--------------------------------------------------------------------------------
--- Prime arithmetic
--------------------------------------------------------------------------------
-
--- Reciprocals modulo naturals.
-recipModNatural :: Natural -> Natural -> Natural
-recipModNatural x p = naturalFromInteger $
- recipModInteger (naturalToInteger x) (naturalToInteger p)
-{-# INLINE recipModNatural #-}
diff --git a/src/Data/Field/Galois/Unity.hs b/src/Data/Field/Galois/Unity.hs
--- a/src/Data/Field/Galois/Unity.hs
+++ b/src/Data/Field/Galois/Unity.hs
@@ -17,7 +17,7 @@
import Data.Group (Group(..))
import GHC.Natural (Natural, naturalToInteger)
import GHC.TypeNats (natVal)
-import Test.Tasty.QuickCheck (Arbitrary(..), choose)
+import Test.QuickCheck (Arbitrary(..), choose)
import Text.PrettyPrint.Leijen.Text (Pretty(..))
import Data.Field.Galois.Base (GaloisField(..))
diff --git a/test/Test/Galois.hs b/test/Test/Galois.hs
--- a/test/Test/Galois.hs
+++ b/test/Test/Galois.hs
@@ -2,7 +2,7 @@
import Protolude
-import Data.Field.Galois hiding (recip)
+import Data.Field.Galois
import Test.Tasty
import Test.Tasty.QuickCheck