/usr/share/axiom-20170501/src/algebra/DDFACT.spad is in axiom-source 20170501-3.
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 | )abbrev package DDFACT DistinctDegreeFactorize
++ Author: P. Gianni, B.Trager
++ Date Created: 1983
++ Date Last Updated: 22 November 1993
++ Description:
++ Package for the factorization of a univariate polynomial with
++ coefficients in a finite field. The algorithm used is the
++ "distinct degree" algorithm of Cantor-Zassenhaus, modified
++ to use trace instead of the norm and a table for computing
++ Frobenius as suggested by Naudin and Quitte .
DistinctDegreeFactorize(F,FP) : SIG == CODE where
F : FiniteFieldCategory
FP : UnivariatePolynomialCategory(F)
fUnion ==> Union("nil", "sqfr", "irred", "prime")
FFE ==> Record(flg:fUnion, fctr:FP, xpnt:Integer)
NNI == NonNegativeInteger
Z == Integer
fact == Record(deg : NNI,prod : FP)
ParFact == Record(irr:FP,pow:Z)
FinalFact == Record(cont:F,factors:List(ParFact))
SIG ==> with
factor : FP -> Factored FP
++ factor(p) produces the complete factorization of the polynomial p.
factorSquareFree : FP -> Factored FP
++ factorSquareFree(p) produces the complete factorization of the
++ square free polynomial p.
distdfact : (FP,Boolean) -> FinalFact
++ distdfact(p,sqfrflag) produces the complete factorization
++ of the polynomial p returning an internal data structure.
++ If argument sqfrflag is true, the polynomial is assumed square free.
separateDegrees : FP -> List fact
++ separateDegrees(p) splits the square free polynomial p into
++ factors each of which is a product of irreducibles of the
++ same degree.
separateFactors : List fact -> List FP
++ separateFactors(lfact) takes the list produced by separateDegrees
++ and produces the complete list of factors.
exptMod : (FP,NNI,FP) -> FP
++ exptMod(u,k,v) raises the polynomial u to the kth power
++ modulo the polynomial v.
trace2PowMod : (FP,NNI,FP) -> FP
++ trace2PowMod(u,k,v) produces the sum of u**(2**i) for i running
++ from 1 to k all computed modulo the polynomial v.
tracePowMod : (FP,NNI,FP) -> FP
++ tracePowMod(u,k,v) produces the sum of \spad{u**(q**i)}
++ for i running and q= size F
irreducible? : FP -> Boolean
++ irreducible?(p) tests whether the polynomial p is irreducible.
CODE ==> add
--declarations
D:=ModMonic(F,FP)
import UnivariatePolynomialSquareFree(F,FP)
--local functions
notSqFr : (FP,FP -> List(FP)) -> List(ParFact)
ddffact : FP -> List(FP)
ddffact1 : (FP,Boolean) -> List fact
ranpol : NNI -> FP
charF : Boolean := characteristic()$F = 2
--construct a random polynomial of random degree < d
ranpol(d:NNI):FP ==
k1: NNI := 0
while k1 = 0 repeat k1 := random d
-- characteristic F = 2
charF =>
u:=0$FP
for j in 1..k1 repeat u:=u+monomial(random()$F,j)
u
u := monomial(1,k1)
for j in 0..k1-1 repeat u:=u+monomial(random()$F,j)
u
notSqFr(m:FP,appl: FP->List(FP)):List(ParFact) ==
factlist : List(ParFact) :=empty()
llf : List FFE
fln :List(FP) := empty()
if (lcm:=leadingCoefficient m)^=1 then m:=(inv lcm)*m
llf:= factorList(squareFree(m))
for lf in llf repeat
d1:= lf.xpnt
pol := lf.fctr
if (lcp:=leadingCoefficient pol)^=1 then pol := (inv lcp)*pol
degree pol=1 => factlist:=cons([pol,d1]$ParFact,factlist)
fln := appl(pol)
factlist :=append([[pf,d1]$ParFact for pf in fln],factlist)
factlist
-- compute u**k mod v (requires call to setPoly of multiple of v)
-- characteristic not equal 2
exptMod(u:FP,k:NNI,v:FP):FP == (reduce(u)$D**k):FP rem v
-- compute u**k mod v (requires call to setPoly of multiple of v)
-- characteristic equal 2
trace2PowMod(u:FP,k:NNI,v:FP):FP ==
uu:=u
for i in 1..k repeat uu:=(u+uu*uu) rem v
uu
-- compute u+u**q+..+u**(q**k) mod v
-- (requires call to setPoly of multiple of v) where q=size< F
tracePowMod(u:FP,k:NNI,v:FP):FP ==
u1 :D :=reduce(u)$D
uu : D := u1
for i in 1..k repeat uu:=(u1+frobenius uu)
(lift uu) rem v
-- compute u**(1+q+..+q**k) rem v where q=#F
-- (requires call to setPoly of multiple of v)
-- frobenius map is used
normPowMod(u:FP,k:NNI,v:FP):FP ==
u1 :D :=reduce(u)$D
uu : D := u1
for i in 1..k repeat uu:=(u1*frobenius uu)
(lift uu) rem v
--find the factorization of m as product of factors each containing
--terms of equal degree .
-- if testirr=true the function returns the first factor found
ddffact1(m:FP,testirr:Boolean):List(fact) ==
p:=size$F
dg:NNI :=0
ddfact:List(fact):=empty()
--evaluation of x**p mod m
k1:NNI
u:= m
du := degree u
setPoly u
mon: FP := monomial(1,1)
v := mon
for k1 in 1.. while k1 <= (du quo 2) repeat
v := lift frobenius reduce(v)$D
g := gcd(v-mon,u)
dg := degree g
dg =0 => "next k1"
if leadingCoefficient g ^=1 then g := (inv leadingCoefficient g)*g
ddfact := cons([k1,g]$fact,ddfact)
testirr => return ddfact
u := u quo g
du := degree u
du = 0 => return ddfact
setPoly u
cons([du,u]$fact,ddfact)
-- test irreducibility
irreducible?(m:FP):Boolean ==
mf:fact:=first ddffact1(m,true)
degree m = mf.deg
--export ddfact1
separateDegrees(m:FP):List(fact) == ddffact1(m,false)
--find the complete factorization of m, using the result of ddfact1
separateFactors(distf : List fact) :List FP ==
ddfact := distf
n1:Integer
p1:=size()$F
if charF then n1:=length(p1)-1
newaux,aux,ris : List FP
ris := empty()
t,fprod : FP
for ffprod in ddfact repeat
fprod := ffprod.prod
d := ffprod.deg
degree fprod = d => ris := cons(fprod,ris)
aux:=[fprod]
setPoly fprod
while ^(empty? aux) repeat
t := ranpol(2*d)
if charF then t:=trace2PowMod(t,(n1*d-1)::NNI,fprod)
else t:=exptMod(tracePowMod(t,(d-1)::NNI,fprod),
(p1 quo 2)::NNI,fprod)-1$FP
newaux:=empty()
for u in aux repeat
g := gcd(u,t)
dg:= degree g
dg=0 or dg = degree u => newaux:=cons(u,newaux)
v := u quo g
if dg=d then ris := cons(inv(leadingCoefficient g)*g,ris)
else newaux := cons(g,newaux)
if degree v=d then ris := cons(inv(leadingCoefficient v)*v,ris)
else newaux := cons(v,newaux)
aux:=newaux
ris
--distinct degree algorithm for monic ,square-free polynomial
ddffact(m:FP):List(FP)==
ddfact:=ddffact1(m,false)
empty? ddfact => [m]
separateFactors ddfact
--factorize a general polynomial with distinct degree algorithm
--if test=true no check is executed on square-free
distdfact(m:FP,test:Boolean):FinalFact ==
factlist: List(ParFact):= empty()
fln : List(FP) :=empty()
--make m monic
if (lcm := leadingCoefficient m) ^=1 then m := (inv lcm)*m
--is x**d factor of m?
if (d := minimumDegree m)>0 then
m := (monicDivide (m,monomial(1,d))).quotient
factlist := [[monomial(1,1),d]$ParFact]
d:=degree m
--is m constant?
d=0 => [lcm,factlist]$FinalFact
--is m linear?
d=1 => [lcm,cons([m,d]$ParFact,factlist)]$FinalFact
--m is square-free
test =>
fln := ddffact m
factlist := append([[pol,1]$ParFact for pol in fln],factlist)
[lcm,factlist]$FinalFact
--factorize the monic,square-free terms
factlist:= append(notSqFr(m,ddffact),factlist)
[lcm,factlist]$FinalFact
--factorize the polynomial m
factor(m:FP) ==
m = 0 => 0
flist := distdfact(m,false)
makeFR(flist.cont::FP,[["prime",u.irr,u.pow]$FFE
for u in flist.factors])
--factorize the square free polynomial m
factorSquareFree(m:FP) ==
m = 0 => 0
flist := distdfact(m,true)
makeFR(flist.cont::FP,[["prime",u.irr,u.pow]$FFE
for u in flist.factors])
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