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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 | )abbrev domain FFCGX FiniteFieldCyclicGroupExtension
++ Authors: J.Grabmeier, A.Scheerhorn
++ Date Created: 04.04.1991
++ References:
++ Grab92 Finite Fields in Axiom
++ Lidl83 Finite Field, Encyclopedia of Mathematics and Its Applications
++ Description:
++ FiniteFieldCyclicGroupExtension(GF,n) implements a extension of degree n
++ over the ground field GF. Its elements are represented by powers of
++ a primitive element, a generator of the multiplicative (cyclic) group.
++ As primitive element we choose the root of the extension polynomial, which
++ is created by createPrimitivePoly from
++ \spadtype{FiniteFieldPolynomialPackage}. Zech logarithms are stored
++ in a table of size half of the field size, and use \spadtype{SingleInteger}
++ for representing field elements, hence, there are restrictions
++ on the size of the field.
FiniteFieldCyclicGroupExtension(GF,extdeg) : SIG == CODE where
GF : FiniteFieldCategory
extdeg : PositiveInteger
PI ==> PositiveInteger
FFPOLY ==> FiniteFieldPolynomialPackage(GF)
SI ==> SingleInteger
SIG ==> FiniteAlgebraicExtensionField(GF) with
getZechTable : () -> PrimitiveArray(SingleInteger)
++ getZechTable() returns the zech logarithm table of the field.
++ This table is used to perform additions in the field quickly.
CODE ==> FiniteFieldCyclicGroupExtensionByPolynomial(GF,_
createPrimitivePoly(extdeg)$FFPOLY)
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