/usr/share/axiom-20170501/src/algebra/FFFACTSE.spad is in axiom-source 20170501-3.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
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 | )abbrev package FFFACTSE FiniteFieldFactorizationWithSizeParseBySideEffect
++ Author: Patrice Naudin, Claude Quitte, Kaj Laursen
++ Date Created: September 1996
++ Date Last Updated: April 2010 by Tim Daly
++ References:
++ Grab92 Finite Fields in Axiom
++ Description:
++ Part of the package for Algebraic Function Fields in one variable (PAFF)
++ It has been modified (very slitely) so that each time the "factor"
++ function is used, the variable related to the size of the field
++ over which the polynomial is factorized is reset. This is done in
++ order to be used with a "dynamic extension field" which size is not
++ fixed but set before calling the "factor" function and which is
++ parse by side effect to this package via the function "size". See
++ the local function "initialize" of this package.
FiniteFieldFactorizationWithSizeParseBySideEffect(K,PolK) : SIG == CODE where
K : FiniteFieldCategory
PolK : UnivariatePolynomialCategory(K)
SIG ==> with
factorSquareFree : PolK -> List(PolK)
factorCantorZassenhaus : (PolK, NonNegativeInteger) -> List(PolK)
factor : PolK -> Factored(PolK)
factorUsingYun : PolK -> Factored(PolK)
factorUsingMusser : PolK -> Factored(PolK)
irreducible? : PolK -> Boolean
CODE ==> add
import FiniteFieldSquareFreeDecomposition(K, PolK)
p : NonNegativeInteger := characteristic()$K
p' : NonNegativeInteger := p quo 2 -- used for odd p : (p-1)/2
q : NonNegativeInteger := size()$K
q' : NonNegativeInteger := q quo 2 -- used for odd q : (q-1)/2
X : PolK := monomial(1, 1)
primeKdim : NonNegativeInteger :=
q_quo_p : NonNegativeInteger := q quo p ; e : NonNegativeInteger := 1
while q_quo_p > 1 repeat (e := e + 1 ; q_quo_p := q_quo_p quo p)
e
initialize(): Void() ==
q : NonNegativeInteger := size()$K
q' : NonNegativeInteger := q quo 2 -- used for odd q : (q-1)/2
primeKdim : NonNegativeInteger :=
q_quo_p : NonNegativeInteger := q quo p ; e:NonNegativeInteger := 1
while q_quo_p > 1 repeat (e := e + 1 ; q_quo_p := q_quo_p quo p)
e
exp(P : PolK, n : NonNegativeInteger, R : PolK) : PolK ==
PP : PolK := P rem R ; Q : PolK := 1
repeat
if odd?(n) then Q := Q * PP rem R
(n := n quo 2) = 0 => leave
PP := PP * PP rem R
return Q
pPowers(P : PolK) : PrimitiveArray(PolK) == -- P is monic
n := degree(P)
result : PrimitiveArray(PolK) := new(n, 1)
result(1) := Qi := Q := exp(X, p, P)
for i in 2 .. n-1 repeat (Qi := Qi*Q rem P ; result(i) := Qi)
return result
pExp(Q : PolK, Xpowers : PrimitiveArray(PolK)) : PolK ==
Q' : PolK := 0
while Q ^= 0 repeat
Q':=Q' +primeFrobenius(leadingCoefficient(Q)) * Xpowers(degree(Q))
Q := reductum(Q)
return Q'
pTrace(Q : PolK, d : NonNegativeInteger, P : PolK,
Xpowers : PrimitiveArray(PolK)) : PolK ==
Q : PolK := Q rem P
result : PolK := Q
for i in 1 .. d-1 repeat result := Q + pExp(result, Xpowers)
return result rem P
random(n : NonNegativeInteger) : PolK ==
repeat
if (deg := (random(n)$Integer)::NonNegativeInteger) > 0 then leave
repeat
if (x : K := random()$K) ^= 0 then leave
result : PolK :=
monomial(x, deg) + +/[monomial(random()$K, i) for i in 0 .. deg-1]
return result
internalFactorCZ(P : PolK, -- P monic-squarefree
d:NonNegativeInteger, Xpowers:PrimitiveArray(PolK)) : List(PolK) ==
listOfFactors : List(PolK) := [P]
degree(P) = d => return listOfFactors
result : List(PolK) := []
pDim : NonNegativeInteger := d * primeKdim
Q : PolK := P
repeat
G := pTrace(random(degree(Q)), pDim, Q, Xpowers)
if p > 2 then G := exp(G, p', Q) - 1
Q1 := gcd(G, Q) ; d1 := degree(Q1)
if d1 > 0 and d1 < degree(Q) then
listOfFactors := rest(listOfFactors)
if d1 = d then result := cons(Q1, result)
else listOfFactors := cons(Q1, listOfFactors)
Q1 := Q quo Q1 ; d1 := degree(Q1)
if d1 = d then result := cons(Q1, result)
else listOfFactors := cons(Q1, listOfFactors)
if empty?(listOfFactors) then leave
Q := first(listOfFactors)
return result
internalFactorSquareFree(P:PolK):List(PolK) == -- P is monic-squareFree
degree(P) = 1 => [P]
result : List(PolK) := []
Xpowers : PrimitiveArray(PolK) := pPowers(P)
S : PolK := Xpowers(1)
for j in 1..primeKdim-1 repeat S := pExp(S, Xpowers)
for i in 1 .. repeat -- S = X**(q**i) mod P
if degree(R := gcd(S - X, P)) > 0 then
result := concat(internalFactorCZ(R, i, Xpowers), result)
if degree (P) = degree (R) then return result
P := P quo R
if i >= degree(P) quo 2 then return cons(P, result)
for j in 0 .. degree(P)-1 repeat Xpowers(j):=Xpowers(j) rem P
S := S rem P
else if i >= degree(P) quo 2 then return cons(P, result)
for j in 1 .. primeKdim repeat S := pExp(S, Xpowers)
internalFactor(P:PolK, sqrfree:PolK -> Factored(PolK)) : Factored(PolK) ==
result : Factored(PolK)
if (d := minimumDegree(P)) > 0 then
P := P quo monomial(1, d)
result := primeFactor(X, d)
else
result := 1
degree(P) = 0 => P * result
if (lcP := leadingCoefficient(P)) ^= 1 then P := inv(lcP) * P
degree(P) = 1 => lcP::PolK * primeFactor(P, 1) * result
sqfP : Factored(PolK) := sqrfree(P)
for x in factors(sqfP) repeat
xFactors : List(PolK) := internalFactorSquareFree(x.factor)
result:=result * */[primeFactor(Q, x.exponent) for Q in xFactors]
return lcP::PolK * result
factorUsingYun(P : PolK) : Factored(PolK) == internalFactor(P, Yun)
factorUsingMusser(P : PolK) : Factored(PolK) == internalFactor(P, Musser)
factor(P : PolK) : Factored(PolK) ==
initialize()
factorUsingYun(P)
factorSquareFree(P : PolK) : List(PolK) ==
degree(P) = 0 => []
discriminant(P) = 0 => error("factorSquareFree : non quadratfrei")
if (lcP := leadingCoefficient(P)) ^= 1 then P := inv(lcP) * P
return internalFactorSquareFree(P)
factorCantorZassenhaus(P : PolK, d : NonNegativeInteger) : List(PolK) ==
if (lcP := leadingCoefficient(P)) ^= 1 then P := inv(lcP) * P
degree(P) = 1 => [P]
return internalFactorCZ(P, d, pPowers(P))
qExp(Q : PolK, XqPowers : PrimitiveArray(PolK)) : PolK ==
Q' : PolK := 0
while Q ^= 0 repeat
Q' := Q' + leadingCoefficient(Q) * XqPowers(degree(Q))
Q := reductum(Q)
return Q'
qPowers (Xq:PolK, P:PolK) : PrimitiveArray(PolK) == -- Xq = X**q mod P
n := degree(P)
result : PrimitiveArray(PolK) := new(n, 1)
result(1) := Q := Xq
for i in 2 .. n-1 repeat (Q := Q*Xq rem P ; result(i) := Q)
return result
discriminantTest?(P : PolK) : Boolean ==
(delta : K := discriminant(P)) = 0 => true
StickelbergerTest : Boolean := (delta ** q' = 1) = even?(degree(P))
return StickelbergerTest
evenCharacteristicIrreducible?(P : PolK) : Boolean ==
(n := degree(P)) = 0 => false
n = 1 => true
degree(gcd(P, D(P))) > 0 => false
if (lcP := leadingCoefficient(P)) ^= 1 then P := inv(lcP) * P
S : PolK := exp(X, q, P)
if degree(gcd(S - X, P)) > 0 then
return false
if n < 4 then return true
maxDegreeToTest : NonNegativeInteger := n quo 2
XqPowers : PrimitiveArray(PolK) := qPowers(S, P)
for i in 2 .. maxDegreeToTest repeat
S := qExp(S, XqPowers)
if degree(gcd(S - X, P)) > 0 then
return false
return true
oddCharacteristicIrreducible?(P : PolK) : Boolean ==
(n := degree(P)) = 0 => false
n = 1 => true
discriminantTest?(P) => false
if (lcP := leadingCoefficient(P)) ^= 1 then P := inv(lcP) * P
S : PolK := exp(X, q, P)
if degree(gcd(S - X, P)) > 0 then
return false
if n < 6 then return true
maxDegreeToTest : NonNegativeInteger := n quo 3
XqPowers : PrimitiveArray(PolK) := qPowers(S, P)
for i in 2 .. maxDegreeToTest repeat
S := qExp(S, XqPowers)
if degree(gcd(S - X, P)) > 0 then
return false
return true
if p = 2 then
irreducible?(P : PolK) : Boolean == evenCharacteristicIrreducible?(P)
else
irreducible?(P : PolK) : Boolean == oddCharacteristicIrreducible?(P)
|