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
|
\documentclass{article}
\usepackage{open-axiom}
\begin{document}
\title{\$SPAD/src/algebra ffx.spad}
\author{Robert Sutor}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{package IRREDFFX IrredPolyOverFiniteField}
<<package IRREDFFX IrredPolyOverFiniteField>>=
)abbrev package IRREDFFX IrredPolyOverFiniteField
++ Author: Robert S. Sutor (original)
++ Date Created: ???
++ Date Last Updated: 29 May 1990
++ Description:
++ This package exports the function generateIrredPoly that computes
++ a monic irreducible polynomial of degree n over a finite field.
IrredPolyOverFiniteField(GF:FiniteFieldCategory): Exports == Impl where
N ==> PositiveInteger
Z ==> Integer
SUP ==> SparseUnivariatePolynomial GF
QR ==> Record(quotient: Z, remainder: Z)
Exports ==> with
generateIrredPoly: N -> SUP
++ generateIrredPoly(n) generates an irreducible univariate
++ polynomial of the given degree n over the finite field.
Impl ==> add
import DistinctDegreeFactorize(GF, SUP)
getIrredPoly : (Z, N) -> SUP
qAdicExpansion: Z -> SUP
p := characteristic$GF :: N
q := size()$GF :: N
qAdicExpansion(z : Z): SUP ==
-- expands z as a sum of powers of q, with coefficients in GF
-- z = HornerEval(qAdicExpansion z,q)
qr := divide(z, q)
zero?(qr.remainder) => monomial(1, 1) * qAdicExpansion(qr.quotient)
r := index(qr.remainder pretend N)$GF :: SUP
zero?(qr.quotient) => r
r + monomial(1, 1) * qAdicExpansion(qr.quotient)
getIrredPoly(start : Z, n : N) : SUP ==
-- idea is to iterate over possibly irreducible monic polynomials
-- until we find an irreducible one. The obviously reducible ones
-- are avoided.
mon := monomial(1, n)$SUP
pol: SUP := 0
found: Boolean := false
end: Z := q**n - 1
while not ((end < start) or found) repeat
if gcd(start, p) = 1 then
if irreducible?(pol := mon + qAdicExpansion(start)) then
found := true
start := start + 1
zero? pol => error "no irreducible poly found"
pol
generateIrredPoly(n : N) : SUP ==
-- want same poly every time
one?(n) => monomial(1, 1)$SUP
one?(gcd(p, n)) or (n < q) =>
odd?(n) => getIrredPoly(2, n)
getIrredPoly(1, n)
getIrredPoly(q + 1, n)
@
\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 IRREDFFX IrredPolyOverFiniteField>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
|
