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\documentclass{article}
\usepackage{open-axiom}
\begin{document}
\title{\$SPAD/src/algebra moddfact.spad}
\author{Barry Trager, James Davenport}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{package MDDFACT ModularDistinctDegreeFactorizer}
<<package MDDFACT ModularDistinctDegreeFactorizer>>=
)abbrev package MDDFACT ModularDistinctDegreeFactorizer
++ Author: Barry Trager
++ Date Created:
++ Date Last Updated: 20.9.95 (JHD)
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ This package supports factorization and gcds
++ of univariate polynomials over the integers modulo different
++ primes. The inputs are given as polynomials over the integers
++ with the prime passed explicitly as an extra argument.
ModularDistinctDegreeFactorizer(U):C == T where
U : UnivariatePolynomialCategory(Integer)
I ==> Integer
NNI ==> NonNegativeInteger
PI ==> PositiveInteger
V ==> Vector
L ==> List
DDRecord ==> Record(factor:EMR,degree:I)
UDDRecord ==> Record(factor:U,degree:I)
DDList ==> L DDRecord
UDDList ==> L UDDRecord
C == with
gcd:(U,U,I) -> U
++ gcd(f1,f2,p) computes the gcd of the univariate polynomials
++ f1 and f2 modulo the integer prime p.
linears: (U,I) -> U
++ linears(f,p) returns the product of all the linear factors
++ of f modulo p. Potentially incorrect result if f is not
++ square-free modulo p.
factor:(U,I) -> L U
++ factor(f1,p) returns the list of factors of the univariate
++ polynomial f1 modulo the integer prime p.
++ Error: if f1 is not square-free modulo p.
ddFact:(U,I) -> UDDList
++ ddFact(f,p) computes a distinct degree factorization of the
++ polynomial f modulo the prime p, i.e. such that each factor
++ is a product of irreducibles of the same degrees. The input
++ polynomial f is assumed to be square-free modulo p.
separateFactors:(UDDList,I) -> L U
++ separateFactors(ddl, p) refines the distinct degree factorization
++ produced by \spadfunFrom{ddFact}{ModularDistinctDegreeFactorizer}
++ to give a complete list of factors.
exptMod:(U,I,U,I) -> U
++ exptMod(f,n,g,p) raises the univariate polynomial f to the nth
++ power modulo the polynomial g and the prime p.
T == add
reduction(u:U,p:I):U ==
zero? p => u
map(positiveRemainder(#1,p),u)
merge(p:I,q:I):Union(I,"failed") ==
p = q => p
p = 0 => q
q = 0 => p
"failed"
modInverse(c:I,p:I):I ==
(extendedEuclidean(c,p,1)::Record(coef1:I,coef2:I)).coef1
exactquo(u:U,v:U,p:I):Union(U,"failed") ==
invlcv:=modInverse(leadingCoefficient v,p)
r:=monicDivide(u,reduction(invlcv*v,p))
reduction(r.remainder,p) ~=0 => "failed"
reduction(invlcv*r.quotient,p)
EMR := EuclideanModularRing(Integer,U,Integer,
reduction,merge,exactquo)
probSplit2:(EMR,EMR,I) -> Union(List EMR,"failed")
trace:(EMR,I,EMR) -> EMR
ddfactor:EMR -> L EMR
ddfact:EMR -> DDList
sepFact1:DDRecord -> L EMR
sepfact:DDList -> L EMR
probSplit:(EMR,EMR,I) -> Union(L EMR,"failed")
makeMonic:EMR -> EMR
exptmod:(EMR,I,EMR) -> EMR
lc(u:EMR):I == leadingCoefficient(u::U)
degree(u:EMR):I == degree(u::U)
makeMonic(u) == modInverse(lc(u),modulus(u)) * u
i:I
exptmod(u1,i,u2) ==
negative? i => error("negative exponentiation not allowed for exptMod")
ans:= 1$EMR
while positive? i repeat
if odd?(i) then ans:= (ans * u1) rem u2
i:= i quo 2
u1:= (u1 * u1) rem u2
ans
exptMod(a,i,b,q) ==
ans:= exptmod(reduce(a,q),i,reduce(b,q))
ans::U
ddfactor(u) ==
if not one?(c:= lc(u)) then u:= makeMonic(u)
ans:= sepfact(ddfact(u))
cons(c::EMR,[makeMonic(f) for f in ans | positive? degree(f)])
gcd(u,v,q) == gcd(reduce(u,q),reduce(v,q))::U
factor(u,q) ==
v:= reduce(u,q)
dv:= reduce(differentiate(u),q)
positive? degree gcd(v,dv) =>
error("Modular factor: polynomial must be squarefree")
ans:= ddfactor v
[f::U for f in ans]
ddfact(u) ==
p:=modulus u
w:= reduce(monomial(1,1)$U,p)
m:= w
d:I:= 1
if not one?(c:= lc(u)) then u:= makeMonic u
ans:DDList:= []
repeat
w:= exptmod(w,p,u)
g:= gcd(w - m,u)
if positive? degree g then
g:= makeMonic(g)
ans:= [[g,d],:ans]
u:= (u quo g)
degree(u) = 0 => return [[c::EMR,0$I],:ans]
d:= d+1
d > (degree(u):I quo 2) =>
return [[c::EMR,0$I],[u,degree(u)],:ans]
ddFact(u,q) ==
ans:= ddfact(reduce(u,q))
[[(dd.factor)::U,dd.degree]$UDDRecord for dd in ans]$UDDList
linears(u,q) ==
uu:=reduce(u,q)
m:= reduce(monomial(1,1)$U,q)
gcd(exptmod(m,q,uu)-m,uu)::U
sepfact(factList) ==
"append"/[sepFact1(f) for f in factList]
separateFactors(uddList,q) ==
ans:= sepfact [[reduce(udd.factor,q),udd.degree]$DDRecord for
udd in uddList]$DDList
[f::U for f in ans]
decode(s:Integer, p:Integer, x:U):U ==
s<p => s::U
qr := divide(s,p)
qr.remainder :: U + x*decode(qr.quotient, p, x)
sepFact1(f) ==
u:= f.factor
p:=modulus u
(d := f.degree) = 0 => [u]
if not one?(c:= lc(u)) then u:= makeMonic(u)
d = (du := degree(u)) => [u]
ans:L EMR:= []
x:U:= monomial(1,1)
-- for small primes find linear factors by exhaustion
d=1 and p < 1000 =>
for i in 0.. while positive? du repeat
if u(i::U) = 0 then
ans := cons(reduce(x-(i::U),p),ans)
du := du-1
ans
y:= x
s:I:= 0
ss:I := 1
stack:L EMR:= [u]
until null stack repeat
t:= reduce(((s::U)+x),p)
if not ((flist:= probSplit(first stack,t,d)) case "failed") then
stack:= rest stack
for fact in flist repeat
f1:= makeMonic(fact)
(df1:= degree(f1)) = 0 => nil
df1 > d => stack:= [f1,:stack]
ans:= [f1,:ans]
p = 2 =>
ss:= ss + 1
x := y * decode(ss, p, y)
s:= s+1
s = p =>
s:= 0
ss := ss + 1
x:= y * decode(ss, p, y)
not one? leadingCoefficient(x) =>
ss := p ** degree x
x:= y ** (degree(x) + 1)
[c * first(ans),:rest(ans)]
probSplit(u,t,d) ==
(p:=modulus(u)) = 2 => probSplit2(u,t,d)
f1:= gcd(u,t)
r:= ((p**(d:NNI)-1) quo 2):NNI
n:= exptmod(t,r,u)
f2:= gcd(u,n + 1)
(g:= f1 * f2) = 1 => "failed"
g = u => "failed"
[f1,f2,(u quo g)]
probSplit2(u,t,d) ==
f:= gcd(u,trace(t,d,u))
f = 1 => "failed"
degree u = degree f => "failed"
[1,f,u quo f]
trace(t,d,u) ==
p:=modulus(t)
d:= d - 1
tt:=t
while positive? d repeat
tt:= (tt + (t:=exptmod(t,p,u))) rem u
d:= d - 1
tt
@
\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 MDDFACT ModularDistinctDegreeFactorizer>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
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