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\documentclass{article}
\usepackage{open-axiom}
\begin{document}
\title{\$SPAD/src/algebra ffpoly2.spad}
\author{Johannes Grabmeier, Alfred Scheerhorn}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{package FFPOLY2 FiniteFieldPolynomialPackage2}
<<package FFPOLY2 FiniteFieldPolynomialPackage2>>=
)abbrev package FFPOLY2 FiniteFieldPolynomialPackage2
++ Authors: J.Grabmeier, A.Scheerhorn
++ Date Created: 26.03.1991
++ Date Last Updated:
++ Basic Operations: rootOfIrreduciblePoly
++ Related Constructors: FiniteFieldCategory
++ Also See:
++ AMS Classifications:
++ Keywords: finite field, zeros of polynomials, Berlekamp's trace algorithm
++ References:
++ R.Lidl, H.Niederreiter: Finite Field, Encycoldia of Mathematics and
++ Its Applications, Vol. 20, Cambridge Univ. Press, 1983, ISBN 0 521 30240 4
++ AXIOM Technical Report Series, to appear.
++ Description:
++ FiniteFieldPolynomialPackage2(F,GF) exports some functions concerning
++ finite fields, which depend on a finite field {\em GF} and an
++ algebraic extension F of {\em GF}, e.g. a zero of a polynomial
++ over {\em GF} in F.
FiniteFieldPolynomialPackage2(F,GF):Exports == Implementation where
F:FieldOfPrimeCharacteristic with
coerce: GF -> F
++ coerce(x) \undocumented{}
lookup: F -> PositiveInteger
++ lookup(x) \undocumented{}
basis: PositiveInteger -> Vector F
++ basis(n) \undocumented{}
Frobenius: F -> F
++ Frobenius(x) \undocumented{}
-- F should be a algebraic extension of the finite field GF, either an
-- algebraic closure of GF or a simple algebraic extension field of GF
GF:FiniteFieldCategory
I ==> Integer
NNI ==> NonNegativeInteger
PI ==> PositiveInteger
SUP ==> SparseUnivariatePolynomial
MM ==> ModMonic(GF,SUP GF)
OUT ==> OutputForm
M ==> Matrix
V ==> Vector
L ==> List
FFPOLY ==> FiniteFieldPolynomialPackage(GF)
SUPF2 ==> SparseUnivariatePolynomialFunctions2(GF,F)
Exports ==> with
rootOfIrreduciblePoly:SUP GF -> F
++ rootOfIrreduciblePoly(f) computes one root of the monic,
++ irreducible polynomial f, which degree must divide the extension degree
++ of {\em F} over {\em GF},
++ i.e. f splits into linear factors over {\em F}.
Implementation ==> add
-- we use berlekamps trace algorithm
-- it is not checked whether the polynomial is irreducible over GF]]
rootOfIrreduciblePoly(pf) ==
-- not irreducible(pf)$FFPOLY =>
-- error("polynomial has to be irreducible")
sizeGF:=size()$GF
-- if the polynomial is of degree one, we're ready
deg:=degree(pf)$(SUP GF)::PI
deg = 0 => error("no roots")
deg = 1 => -coefficient(pf,0)$(SUP GF)::F
p : SUP F := map(coerce,pf)$SUPF2
-- compute qexp, qexp(i) = x **(size()GF ** i) mod p
-- with this list it's easier to compute the gcd(p(x),trace(x))
qexp:=reducedQPowers(pf)$FFPOLY
stillToFactor:=p
-- take linear independent elements, the basis of F over GF
basis:Vector F:=basis(deg)$F
basispointer:I:=1
-- as p is irreducible over GF, 0 can't be a root of p
-- therefore we can use the predicate zero?(root) for indicating
-- whether a root is found
root:=0$F
while zero?(root)$F repeat
beta:F:=basis.basispointer
-- gcd(trace(x)+gf,p(x)) has degree 0,that's why we skip beta=1
if beta = 1$F then
basispointer:=basispointer + 1
beta:= basis.basispointer
basispointer:=basispointer+1
-- compute the polynomial trace(beta * x) mod p(x) using explist
trModp:SUP F:= map(coerce,qexp.0)$SUPF2 * beta
for i in 1..deg-1 repeat
beta:=Frobenius(beta)
trModp:=trModp +$(SUP F) beta *$(SUP F) map(coerce,qexp.i)$SUPF2
-- if it is of degree 0, it doesn't help us finding a root
if positive? degree(trModp)$(SUP F) then
-- for all elements gf of GF do
for j in 1..sizeGF repeat
-- compute gcd(trace(beta * x) + gf,stillToFactor)
h:=gcd(stillToFactor,trModp +$(SUP F) _
(index(j pretend PI)$GF::F::(SUP F)))$(SUP F)
-- make the gcd polynomial monic
if not one? leadingCoefficient(h)$(SUP F) then
h:= (inv leadingCoefficient(h)) * h
degh:=degree(h)$(SUP F)
degSTF:=degree(stillToFactor)$(SUP F)
-- if the gcd has degree one we are ready
degh = 1 => root:=-coefficient(h,0)$(SUP F)
-- if the quotient of stillToFactor and the gcd has
-- degree one, we're also ready
degSTF - degh = 1 =>
root:= -coefficient(stillToFactor quo h,0)$(SUP F)
-- otherwise the gcd helps us finding a root, only if its
-- degree is between 2 and degree(stillToFactor)-2
if degh > 1 and degh < degSTF then
2*degh > degSTF => stillToFactor := stillToFactor quo h
stillToFactor := h
root
@
\section{License}
<<license>>=
--Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
--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 The Numerical ALgorithms Group Ltd. nor the
-- names of its 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.
@
<<*>>=
<<license>>
<<package FFPOLY2 FiniteFieldPolynomialPackage2>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
|
