From ab8cc85adde879fb963c94d15675783f2cf4b183 Mon Sep 17 00:00:00 2001
From: dos-reis <gdr@axiomatics.org>
Date: Tue, 14 Aug 2007 05:14:52 +0000
Subject: Initial population.

---
 src/algebra/ffx.spad.pamphlet | 120 ++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 120 insertions(+)
 create mode 100644 src/algebra/ffx.spad.pamphlet

(limited to 'src/algebra/ffx.spad.pamphlet')

diff --git a/src/algebra/ffx.spad.pamphlet b/src/algebra/ffx.spad.pamphlet
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+++ b/src/algebra/ffx.spad.pamphlet
@@ -0,0 +1,120 @@
+\documentclass{article}
+\usepackage{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)$SUP
+      (n = 1) => monomial(1, 1)$SUP
+--      one?(gcd(p, n)) or (n < q) =>
+      (gcd(p, n) = 1) 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}
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