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--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.
)abbrev domain FFP FiniteFieldExtensionByPolynomial
++ Authors: R.Sutor, J. Grabmeier, O. Gschnitzer, A. Scheerhorn
++ Date Created:
++ Date Last Updated: May 29, 2009
++ Basic Operations:
++ Related Constructors:
++ Also See: FiniteFieldCyclicGroupExtensionByPolynomial,
++ FiniteFieldNormalBasisExtensionByPolynomial
++ AMS Classifications:
++ Keywords: field, extension field, algebraic extension,
++ finite extension, finite field, Galois field
++ Reference:
++ R.Lidl, H.Niederreiter: Finite Field, Encyclopedia of Mathematics an
++ Its Applications, Vol. 20, Cambridge Univ. Press, 1983, ISBN 0 521 30240 4
++ J. Grabmeier, A. Scheerhorn: Finite Fields in AXIOM.
++ AXIOM Technical Report Series, ATR/5 NP2522.
++ Description:
++ FiniteFieldExtensionByPolynomial(GF, defpol) implements the extension
++ of the finite field {\em GF} generated by the extension polynomial
++ {\em defpol} which MUST be irreducible.
++ Note: the user has the responsibility to ensure that
++ {\em defpol} is irreducible.
FiniteFieldExtensionByPolynomial(GF:FiniteFieldCategory,_
defpol:SparseUnivariatePolynomial GF): Exports == Implementation where
-- GF : FiniteFieldCategory
-- defpol : SparseUnivariatePolynomial GF
PI ==> PositiveInteger
NNI ==> NonNegativeInteger
SUP ==> SparseUnivariatePolynomial
I ==> Integer
R ==> Record(key:PI,entry:NNI)
TBL ==> Table(PI,NNI)
SAE ==> SimpleAlgebraicExtension(GF,SUP GF,defpol)
OUT ==> OutputForm
Exports ==> FiniteAlgebraicExtensionField(GF)
Implementation ==> add
-- global variables ====================================================
Rep:=SAE
extdeg:PI := degree(defpol)$(SUP GF) pretend PI
-- the extension degree
alpha := new()$Symbol :: OutputForm
-- a new symbol for the output form of field elements
sizeCG:Integer := size()$GF**extdeg - 1
-- the order of the multiplicative group
facOfGroupSize := nil()$(List Record(factor:Integer,exponent:Integer))
-- the factorization of sizeCG
normalElt:PI:=1
-- for the lookup of the normal Element computed by
-- createNormalElement
primitiveElt:PI:=1
-- for the lookup of the primitive Element computed by
-- createPrimitiveElement()
initlog?:Boolean:=true
-- gets false after initialization of the discrete logarithm table
initelt?:Boolean:=true
-- gets false after initialization of the primitive and the
-- normal element
discLogTable:Table(PI,TBL):=table()$Table(PI,TBL)
-- tables indexed by the factors of sizeCG,
-- discLogTable(factor) is a table with keys
-- primitiveElement() ** (i * (sizeCG quo factor)) and entries i for
-- i in 0..n-1, n computed in initialize() in order to use
-- the minimal size limit 'limit' optimal.
-- functions ===========================================================
-- createNormalElement() ==
-- a:=primitiveElement()
-- nElt:=generator()
-- for i in 1.. repeat
-- normal? nElt => return nElt
-- nElt:=nElt*a
-- nElt
generator() == reduce(monomial(1,1)$SUP(GF))$Rep
norm x == resultant(defpol, lift x)
initializeElt: () -> Void
initializeLog: () -> Void
basis(n:PI) ==
(extdeg rem n) ~= 0 => error "argument must divide extension degree"
a:$:=norm(primitiveElement(),n)
vector [a**i for i in 0..n-1]
degree(x: %): PositiveInteger ==
y:$:=1
m:=zero(extdeg,extdeg+1)$(Matrix GF)
for i in 1..extdeg+1 repeat
setColumn!(m,i,coordinates(y))$(Matrix GF)
y:=y*x
rank(m)::PI
minimalPolynomial(x:$) ==
y:$:=1
m:=zero(extdeg,extdeg+1)$(Matrix GF)
for i in 1..extdeg+1 repeat
setColumn!(m,i,coordinates(y))$(Matrix GF)
y:=y*x
v:=first nullSpace(m)$(Matrix GF)
+/[monomial(v.(i+1),i)$(SUP GF) for i in 0..extdeg]
normal?(x) ==
l:List List GF:=[entries coordinates x]
a:=x
for i in 2..extdeg repeat
a:=Frobenius(a)
l:=concat(l,entries coordinates a)$(List List GF)
((rank matrix(l)$(Matrix GF)) = extdeg::NNI) => true
false
a:GF * x:$ == a *$Rep x
n:I * x:$ == n *$Rep x
-x == -$Rep x
random() == random()$Rep
coordinates(x:$) == coordinates(x)$Rep
represents(v) == represents(v)$Rep
coerce(x:GF):$ == coerce(x)$Rep
definingPolynomial() == defpol
retract(x) == retract(x)$Rep
retractIfCan(x) == retractIfCan(x)$Rep
index(x) == index(x)$Rep
lookup(x) == lookup(x)$Rep
x:$/y:$ == x /$Rep y
x:$/a:GF == x/coerce(a)
-- x:$ / a:GF ==
-- a = 0$GF => error "division by zero"
-- x * inv(coerce(a))
x:$ * y:$ == x *$Rep y
x:$ + y:$ == x +$Rep y
x:$ - y:$ == x -$Rep y
x:$ = y:$ == x =$Rep y
basis() == basis()$Rep
0 == 0$Rep
1 == 1$Rep
factorsOfCyclicGroupSize() ==
if empty? facOfGroupSize then initializeElt()
facOfGroupSize
representationType() == "polynomial"
tableForDiscreteLogarithm(fac) ==
if initlog? then initializeLog()
tbl:=search(fac::PI,discLogTable)$Table(PI,TBL)
tbl case "failed" =>
error "tableForDiscreteLogarithm: argument must be prime divisor_
of the order of the multiplicative group"
tbl pretend TBL
primitiveElement() ==
if initelt? then initializeElt()
index(primitiveElt)
normalElement() ==
if initelt? then initializeElt()
index(normalElt)
initializeElt() ==
facOfGroupSize:=factors(factor(sizeCG)$Integer)
-- get a primitive element
pE:=createPrimitiveElement()
primitiveElt:=lookup(pE)
-- create a normal element
nElt:=generator()
while not normal? nElt repeat
nElt:=nElt*pE
normalElt:=lookup(nElt)
-- set elements initialization flag
initelt? := false
initializeLog() ==
if initelt? then initializeElt()
-- set up tables for discrete logarithm
limit:Integer:=30
-- the minimum size for the discrete logarithm table
for f in facOfGroupSize repeat
fac:=f.factor
base:$:=primitiveElement() ** (sizeCG quo fac)
l:Integer:=length(fac)$Integer
n:Integer:=0
if odd?(l)$Integer then n:=shift(fac,-(l quo 2))
else n:=shift(1,(l quo 2))
if n < limit then
d:=(fac-1) quo limit + 1
n:=(fac-1) quo d + 1
tbl:TBL:=table()$TBL
a:$:=1
for i in (0::NNI)..(n-1)::NNI repeat
insert!([lookup(a),i::NNI]$R,tbl)$TBL
a:=a*base
insert!([fac::PI,copy(tbl)$TBL]_
$Record(key:PI,entry:TBL),discLogTable)$Table(PI,TBL)
-- set logarithm initialization flag
initlog? := false
-- tell user about initialization
--print("discrete logarithm tables initialized"::OUT)
coerce(e:$):OutputForm == outputForm(lift(e),alpha)
extensionDegree(): PositiveInteger == extdeg
size() == (sizeCG + 1) pretend NNI
-- sizeOfGroundField() == size()$GF
inGroundField?(x) ==
retractIfCan(x) = "failed" => false
true
characteristic == characteristic$GF
before?(x,y) == before?(x,y)$Rep
)abbrev domain FFX FiniteFieldExtension
++ Authors: R.Sutor, J. Grabmeier, A. Scheerhorn
++ Date Created:
++ Date Last Updated: 31 March 1991
++ Basic Operations:
++ Related Constructors: FiniteFieldExtensionByPolynomial,
++ FiniteFieldPolynomialPackage
++ Also See: FiniteFieldCyclicGroupExtension,
++ FiniteFieldNormalBasisExtension
++ AMS Classifications:
++ Keywords: field, extension field, algebraic extension,
++ finite extension, finite field, Galois field
++ Reference:
++ R.Lidl, H.Niederreiter: Finite Field, Encyclopedia of Mathematics an
++ Its Applications, Vol. 20, Cambridge Univ. Press, 1983, ISBN 0 521 30240 4
++ J. Grabmeier, A. Scheerhorn: Finite Fields in AXIOM.
++ AXIOM Technical Report Series, ATR/5 NP2522.
++ Description:
++ FiniteFieldExtensionByPolynomial(GF, n) implements an extension
++ of the finite field {\em GF} of degree n generated by the extension
++ polynomial constructed by
++ \spadfunFrom{createIrreduciblePoly}{FiniteFieldPolynomialPackage} from
++ \spadtype{FiniteFieldPolynomialPackage}.
FiniteFieldExtension(GF, n): Exports == Implementation where
GF: FiniteFieldCategory
n : PositiveInteger
Exports ==> FiniteAlgebraicExtensionField(GF)
-- MonogenicAlgebra(GF, SUP) with -- have to check this
Implementation ==> FiniteFieldExtensionByPolynomial(GF,
createIrreduciblePoly(n)$FiniteFieldPolynomialPackage(GF))
-- old code for generating irreducible polynomials:
-- now "better" order (sparse polys first)
-- generateIrredPoly(n)$IrredPolyOverFiniteField(GF))
)abbrev domain IFF InnerFiniteField
++ Author: ???
++ Date Created: ???
++ Date Last Updated: 29 May 1990
++ Basic Operations:
++ Related Constructors: FiniteFieldExtensionByPolynomial,
++ FiniteFieldPolynomialPackage
++ Also See: FiniteFieldCyclicGroup, FiniteFieldNormalBasis
++ AMS Classifications:
++ Keywords: field, extension field, algebraic extension,
++ finite extension, finite field, Galois field
++ Reference:
++ R.Lidl, H.Niederreiter: Finite Field, Encyclopedia of Mathematics an
++ Its Applications, Vol. 20, Cambridge Univ. Press, 1983, ISBN 0 521 30240 4
++ J. Grabmeier, A. Scheerhorn: Finite Fields in AXIOM.
++ AXIOM Technical Report Series, ATR/5 NP2522.
++ Description:
++ InnerFiniteField(p,n) implements finite fields with \spad{p**n} elements
++ where p is assumed prime but does not check.
++ For a version which checks that p is prime, see \spadtype{FiniteField}.
InnerFiniteField(p:PositiveInteger, n:PositiveInteger) ==
FiniteFieldExtension(InnerPrimeField p, n)
)abbrev domain FF FiniteField
++ Author: ???
++ Date Created: ???
++ Date Last Updated: 29 May 1990
++ Basic Operations:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords: field, extension field, algebraic extension,
++ finite extension, finite field, Galois field
++ Reference:
++ R.Lidl, H.Niederreiter: Finite Field, Encyclopedia of Mathematics an
++ Its Applications, Vol. 20, Cambridge Univ. Press, 1983, ISBN 0 521 30240 4
++ J. Grabmeier, A. Scheerhorn: Finite Fields in AXIOM.
++ AXIOM Technical Report Series, ATR/5 NP2522.
++ Description:
++ FiniteField(p,n) implements finite fields with p**n elements.
++ This packages checks that p is prime.
++ For a non-checking version, see \spadtype{InnerFiniteField}.
FiniteField(p:PositiveInteger, n:PositiveInteger): _
FiniteAlgebraicExtensionField(PrimeField p) ==_
FiniteFieldExtensionByPolynomial(PrimeField p,_
createIrreduciblePoly(n)$FiniteFieldPolynomialPackage(PrimeField p))
-- old code for generating irreducible polynomials:
-- now "better" order (sparse polys first)
-- generateIrredPoly(n)$IrredPolyOverFiniteField(GF))
|