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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 | )abbrev domain FFCG FiniteFieldCyclicGroup
++ 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:
++ FiniteFieldCyclicGroup(p,n) implements a finite field extension of degee n
++ over the prime field with p elements. 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}. The 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.
FiniteFieldCyclicGroup(p,extdeg) : SIG == CODE where
p : PositiveInteger
extdeg : PositiveInteger
PI ==> PositiveInteger
FFPOLY ==> FiniteFieldPolynomialPackage(PrimeField(p))
SI ==> SingleInteger
SIG ==> FiniteAlgebraicExtensionField(PrimeField(p)) 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(PrimeField(p),_
createPrimitivePoly(extdeg)$FFPOLY)
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