BerlekampAlgorithm (empty) → 0.1.0.0

raw patch · 4 files changed

+213/−0 lines, 4 filesdep +basedep +besoutsetup-changed

Dependencies added: base, besout

Files

• BerlekampAlgorithm.cabal +54/−0
• BerlekampAlgorithm.hs +127/−0
• LICENSE +30/−0
• Setup.hs +2/−0

@@ -0,0 +1,54 @@+-- Initial BerlekampAlgorithm.cabal generated by cabal init.  For further +-- documentation, see http://haskell.org/cabal/users-guide/++-- The name of the package.+name:                BerlekampAlgorithm++-- The package version.  See the Haskell package versioning policy (PVP) +-- for standards guiding when and how versions should be incremented.+-- http://www.haskell.org/haskellwiki/Package_versioning_policy+-- PVP summary:      +-+------- breaking API changes+--                   | | +----- non-breaking API additions+--                   | | | +--- code changes with no API change+version:             0.1.0.0++-- A short (one-line) description of the package.+synopsis:            Factorization of polynomials over finite field++-- A longer description of the package.+-- description:         ++-- The license under which the package is released.+license:             BSD3++-- The file containing the license text.+license-file:        LICENSE++-- The package author(s).+author:              Abdelwaheb Miled++-- An email address to which users can send suggestions, bug reports, and +-- patches.+maintainer:          abdelwahebmiled@gmail.com++-- A copyright notice.+-- copyright:           ++category:            Math++build-type:          Simple++-- Constraint on the version of Cabal needed to build this package.+cabal-version:       >=1.8+++library+  -- Modules exported by the library.+  exposed-modules:     BerlekampAlgorithm+  +  -- Modules included in this library but not exported.+  -- other-modules:       +  +  -- Other library packages from which modules are imported.+  build-depends:       base ==4.6.*, besout ==0.2.*+  

@@ -0,0 +1,127 @@+module BerlekampAlgorithm ( g,frob, pivotPos',lswap,triangulizedModIntegerMat,nullSpaceModIntegerMat,mmultZ,+matrixBerlTranspose ,derivPolyZ,squareFreePolyZ,irreducibilityTestPolyZ,berlekamp,multPoly) where+import Data.List+import Bezout +-- |Berlekamp's Factorization Algorithm over Fp[x] : computes the factorization of a monic square-free polynomial P into irreducible factor polynomials over F_{p}[x] , p is a prime number. This method is based on linear algebra over finite field.+-- | g+g x = if x == Nothing then (- 1) else (\(Just i)->i) x++-- | pivotPos'+pivotPos' z x = let y = pivotMin x in +		case y of (-1,-1) -> (-1,-1)+			  _  -> pivotPos z x where +				pivotPos _ [[]]  = (-1,-1)+				pivotPos z x = let y = map (head) x in +					let w = (findIndex (/= 0) y) in +					let w' = g $! w in +					if (w' /= (-1)) then (w',z) else pivotPos (z+1) (map (tail) $ x) where +pivotMin x = let y = map (findIndex (/=0)) x in +		let w = (findIndex (/= Nothing) y) in +		let z = (find (/= Nothing) y) in (g w,h z) where +		h x = if x == Nothing then (- 1) else (\(Just (Just i))->i) x++-- | lswap+lswap i j (xs , ys) = (sswap i j xs , sswap i j ys) where +sswap i j xs = if i == j then xs else +		take l xs ++ [xs !! u] ++ (take (u-l-1) $ drop (l+1) xs) ++ [xs !! l] ++ drop (u+1) xs+          where l = if i<j then i else j+                u = if i>j then i else j++-- | triangulizedModIntegerMat+--triangulizedModIntegerMat p m: gives the gauss triangular decomposition of an integeral matrix m in Fp.+-- The result is (r, u) where u is a unimodular matrix, r is an upper-triangular matrix , and u.m = r.++triangulizedModIntegerMat p m = let b = fromIntegral $ length m in hnf' p (m , matI b) where+hnf' :: Integer ->  ([[Integer]],[[Integer]]) -> ([[Integer]],[[Integer]])+hnf' p (a,b) =  let (px , py) = pivotPos' 0 a in+		case (px , py) of (-1,-1) -> (a,b)+			          _ -> let (u ,v) = lswap 0 px (a,b) in+				       let (c:cs , d:ds) = gaussZ' p (u,v) in +				       let (nu , nv) = hnf' p ( cs , ds) in (c:nu , d:nv) where +gaussZ' p (u,v) = let (x,y) = gaussZ p (u,v) in (last x : init x , last y : init y) where +gaussZ :: Integer -> ([[Integer]],[[Integer]]) -> ([[Integer]],[[Integer]])+gaussZ _ ([x], a ) = ([x], a)+gaussZ p ((x:y:xs), (s:t:ix)) = let beta = fromIntegral $ g (findIndex (/= 0) x) in +		let  [a,b] = [x!!beta, y!!beta] in +		let d = (b * (inverseMod a p)) in+		let r = red p d x y in let m = red p d s t in +		let (u , v) = gaussZ p ((x:xs) ,(s:ix)) in (r:u, m:v) where +		red p d x y = zipWith (\c1 c2 -> mods (c2 - (c1 * d)) p ) x y where +matI n = [ [fromIntegral $ fromEnum $ i == j | i <- [1..n]] | j <- [1..n]]++-- | nullSpaceModIntegerMat p m : computes the null space of matrix m in Fp+nullSpaceModIntegerMat :: Integer -> [[Integer]] -> [[Integer]]+nullSpaceModIntegerMat p x = let (y,z) = triangulizedModIntegerMat p x in ff (y,z) where+ff ([],_) = []+ff (x:xs,y:ys) =  if (nub $! x) == [0] then y : ff (xs,ys) else ff (xs,ys)+++-- | mmultZ+-- mmultZ p a b : compute the product of two integer matrices in Fp.+mmultZ p a b = [ [ let y = sum $ zipWith (*) ar bc in mods y p | bc <- (transpose b)] | ar <- a ]+++-- | gcdPolyZ+-- gcdPolyZ p P1 P2 : gives the polynomial gcd of P1 , P2 modulo over Fp[x].+gcdPolyZ p x y = let u = last $ extendedgcdpoly p x y in let v = head u in +			let t = inverseMod v p in mMod (map (* t) u) p++--frob+-- | Frobenius automorphism : linear map V -> V^p - V , V in Fp[x]/P and Fp[x]/P as vector space over the field Fp.++frob :: Integer -> Integer -> [Integer] -> [Integer]+frob _ 0 _ = [0]+frob p k h = let y = prettyFormPoly [[1, p * k],[- 1 , k ],[0,0]] in reverse $! last $! euclideanPolyMod p y h +++-- |matrixBerl+-- matrixBerl p f : is the matrix of the Frobenius endomorphism over the canonical base {1,X,X^2..,X^(p-1)} ,+-- matrixBerl(i,j) = X^(pj)-X^j mod P.+matrixBerlTranspose :: Integer -> [Integer] -> [[Integer]]+matrixBerlTranspose p h = let a = genericLength h - 1 in+	 [let u = fromIntegral k in let v = frob p u h in shift (a - length v) v | k <- [0 .. a - 1]]++-- derivPolyZ+-- | derivPolyZ : derivative of polynmial P over Fp[x]+derivPolyZ :: Integer -> [Integer] -> [Integer]+derivPolyZ _ [] = []+derivPolyZ p x = init $ reverse [let y = reverse x in +		let j = fromIntegral i in mods (j * (y!!i)) p | i <- [0..length x - 1]]++-- | squareFreePolyZ+-- squareFreePolyZ p f : gives the euclidean quotient of P and gcd(f,f'). That quotient is a square free polynomial.+squareFreePolyZ :: Integer -> [Integer] -> [Integer]+squareFreePolyZ p x = let y = derivPolyZ p x in +			let z = trim' $ gcdPolyZ p x y in if z == [1] then x else head $ euclideanPolyMod p x z+++-- |berlekamp+-- berlekamp p P: gives a complete factorization of a polynom P of irreducible polynoms over Fp[x].+berlekamp :: Integer -> [Integer] -> [[Integer]]+berlekamp p f = let g = squareFreePolyZ p f in +	let v = map (trim) $! map (reverse ) $! nullSpaceModIntegerMat p (matrixBerlTranspose p g) in+	let q = length $! v in +	case q of 1 -> [g]+		  _ -> let (a:b:bs) = phi p g v in 			+			let c = zipgcdPoly p a b in if length c == q then c else zipgcdPoly p c (head bs) where +zipgcdPoly _ [] _  = []+zipgcdPoly p (x:xs) y = let z = zgcdPoly p x y  in z ++ zipgcdPoly p xs y where+zgcdPoly _ _ [] = []+zgcdPoly p x (y:ys) = let u = gcdPolyZ p x y in if length u > 1 then u:zgcdPoly p x ys else zgcdPoly p x ys where+phi _ _ [] = []+phi p x (v:vs) = let u = filter (\x -> length x > 1) $! [gcdPolyZ p x ((+:) (-) v [i]) | i <- [0 .. p - 1] ] in u : phi p x vs++-- | irreducibilityTestPolyZ+-- irreducibilityTestPolyZ : irreducibility test of polynomials over Fp[x]+irreducibilityTestPolyZ :: Integer -> [Integer] -> Bool+irreducibilityTestPolyZ p f = let g = squareFreePolyZ p f in if g /= f then False else let u = matrixBerlTranspose p g in+	let v = nullSpaceModIntegerMat p u in+	let q = length $ v in +	case q of 1 -> True+		  _ -> False +-- | multPoly+-- multPoly : product of polynomials P1, .., Pk in Fp[x].+multPoly p x = head $! multPoly' p x where+multPoly' :: Integer ->[[Integer]]-> [[Integer]] +multPoly' _ [x] = [x]+multPoly' p (x:t:xs) = let y = multPolyZ p x t in multPoly' p (y:xs)

@@ -0,0 +1,30 @@+Copyright (c) 2013, Abdelwaheb Miled++All rights reserved.++Redistribution and use in source and binary forms, with or without+modification, are permitted provided that the following conditions are met:++    * Redistributions of source code must retain the above copyright+      notice, this list of conditions and the following disclaimer.++    * Redistributions in binary form must reproduce the above+      copyright notice, this list of conditions and the following+      disclaimer in the documentation and/or other materials provided+      with the distribution.++    * Neither the name of Abdelwaheb Miled nor the names of other+      contributors may be used to endorse or promote products derived+      from this software without specific prior written permission.++THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS+"AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT+LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR+A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT+OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,+SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT+LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,+DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY+THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT+(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE+OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.

@@ -0,0 +1,2 @@+import Distribution.Simple+main = defaultMain