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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.
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-- - 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
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--LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
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--SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
)abbrev domain IPF InnerPrimeField
-- Argument MUST be a prime.
-- This domain does not check, PrimeField does.
++ Authors: N.N., J.Grabmeier, A.Scheerhorn
++ Date Created: ?, November 1990, 26.03.1991
++ Date Last Updated: May 29, 2009
++ Basic Operations:
++ Related Constructors: PrimeField
++ Also See:
++ AMS Classifications:
++ Keywords: prime characteristic, prime field, finite field
++ 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:
++ InnerPrimeField(p) implements the field with p elements.
++ Note: argument p MUST be a prime (this domain does not check).
++ See \spadtype{PrimeField} for a domain that does check.
InnerPrimeField(p:PositiveInteger): Exports == Implementation where
I ==> Integer
NNI ==> NonNegativeInteger
PI ==> PositiveInteger
TBL ==> Table(PI,NNI)
R ==> Record(key:PI,entry:NNI)
SUP ==> SparseUnivariatePolynomial
OUT ==> OutputForm
Exports ==> Join(FiniteFieldCategory,FiniteAlgebraicExtensionField($),_
ConvertibleTo(Integer))
Implementation ==> IntegerMod p add
initializeElt:() -> Void
initializeLog:() -> Void
-- global variables ====================================================
primitiveElt:PI:=1
-- for the lookup the primitive Element computed by createPrimitiveElement()
sizeCG :=(p-1) pretend NonNegativeInteger
-- the size of the cyclic group
facOfGroupSize := nil()$(List Record(factor:Integer,exponent:Integer))
-- the factorization of the cyclic group size
initlog?:Boolean:=true
-- gets false after initialization of the logarithm table
initelt?:Boolean:=true
-- gets false after initialization of the primitive Element
discLogTable:Table(PI,TBL):=table()$Table(PI,TBL)
-- tables indexed by the factors of the size q of the cyclic group
-- discLogTable.factor is a table of with keys
-- primitiveElement() ** (i * (q quo factor)) and entries i for
-- i in 0..n-1, n computed in initialize() in order to use
-- the minimal size limit 'limit' optimal.
-- functions ===========================================================
generator() == 1
-- This uses x**(p-1)=1 (mod p), so x**(q(p-1)+r) = x**r (mod p)
x:$ ** n:Integer ==
zero?(n) => 1
zero?(x) => 0
r := positiveRemainder(n,p-1)::NNI
per (rep(x) **$IntegerMod(p) r)
if p <= convert(max()$SingleInteger)@Integer then
q := p::SingleInteger
recip x ==
zero?(y := convert(x)@Integer :: SingleInteger) => "failed"
invmod(y, q)::Integer::$
else
recip x ==
zero?(y := convert(x)@Integer) => "failed"
invmod(y, p)::$
convert(x:$) == x pretend I
normalElement() == 1
createNormalElement() == 1
characteristic == p
factorsOfCyclicGroupSize() ==
p=2 => facOfGroupSize -- this fixes an infinite loop of functions
-- calls, problem was that factors factor(1)
-- is the empty list
if empty? facOfGroupSize then initializeElt()
facOfGroupSize
representationType() == "prime"
tableForDiscreteLogarithm(fac) ==
if initlog? then initializeLog()
tbl:=search(fac::PI,discLogTable)$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)
initializeElt() ==
facOfGroupSize:=factors(factor(sizeCG)$I)$(Factored I)
-- get a primitive element
primitiveElt:=lookup(createPrimitiveElement())
-- set 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)
-- tell user about initialization
-- print("discrete logarithm table initialized"::OUT)
-- set initialization flag
initlog? := false
degree(x):PI == 1::PositiveInteger
extensionDegree():PI == 1::PositiveInteger
-- sizeOfGroundField() == p::NonNegativeInteger
inGroundField?(x) == true
coordinates(x: %) == new(1,x)$(Vector $)
represents(v) == v.1
retract(x) == x
retractIfCan(x) == x
basis() == new(1,1::$)$(Vector $)
basis(n:PI) ==
n = 1 => basis()
error("basis: argument must divide extension degree")
definingPolynomial() ==
monomial(1,1)$(SUP $) - monomial(1,0)$(SUP $)
minimalPolynomial(x) ==
monomial(1,1)$(SUP $) - monomial(x,0)$(SUP $)
charthRoot(x: %): % == x
before?(x,y) == before?(convert x, convert y)$I
)abbrev domain PF PrimeField
++ Authors: N.N.,
++ Date Created: November 1990, 26.03.1991
++ Date Last Updated: 31 March 1991
++ Basic Operations:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords: prime characteristic, prime field, finite field
++ References:
++ R.Lidl, H.Niederreiter: Finite Field, Encycoldia of Mathematics and
++ Its Applications, Vol. 20, Cambridge Univ. Press, 1983, ISBN 0 521 30240 4
++ Description:
++ PrimeField(p) implements the field with p elements if p is a
++ prime number.
++ Error: if p is not prime.
++ Note: this domain does not check that argument is a prime.
--++ with new compiler, want to put the error check before the add
PrimeField(p:PositiveInteger): Exp == Impl where
Exp ==> Join(FiniteFieldCategory,FiniteAlgebraicExtensionField($),_
ConvertibleTo(Integer))
Impl ==> InnerPrimeField(p) add
if not prime?(p)$IntegerPrimesPackage(Integer) then
error "Argument to prime field must be a prime"
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