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StandardFF

( Version 1.0 )

September 2023

Frank Lübeck
Email: Frank.Luebeck@Math.RWTH-Aachen.De
Homepage: https://www.math.rwth-aachen.de/~Frank.Luebeck

Copyright

This package may be distributed under the terms and conditions of the GNU Public License Version 3 or later, see https://www.gnu.org/licenses.

Colophon

This package implements the constructions in the paper [Lüb23], that is it provides relatively easy to reproduce generators of finite fields and compatible generators of their multiplicative cyclic subgroups.

1 Introduction to StandardFF package

1.1 Aim

2 Standard finite fields

2.1 Definition of standard finite fields

2.2 Creating standard finite fields

2.2-1 Constructing standard finite fields
2.2-2 Filters for standard fields

2.3 Elements in standard finite fields

2.3-1 Maps for elements of standard finite fields

2.4 Embeddings of standard finite fields

  2.4-1 SteinitzPair
  2.4-2 Embedding
  2.4-3 ZZ
  2.4-4 MoveToSmallestStandardField
  2.4-5 StandardIsomorphismGF

3 Standard generators of cyclic groups

3.1 Generators of multiplicative groups

3.1-1 StandardCyclicGenerator

4 Utilities from the StandardFF package

4.1 A simple bijection on a range

4.1-1 StandardAffineShift

4.2 Finding linear combinations

4.2-1 FindLinearCombination

4.3 Irreducibility over finite fields

4.3-1 IsIrreducibleCoeffList

4.4 Connection to Conway polynomials

  4.4-1 FindConjugateZeroes
  4.4-2 ZeroesConway
  4.4-3 SteinitzPairConwayGenerator

4.5 Discrete logarithms

4.5-1 DLog

4.6 Minimal polynomials of sequences

  4.6-1 InvModCoeffs
  4.6-2 BerlekampMassey
  4.6-3 MinimalPolynomialByBerlekampMassey

4.7 Brauer characters with respect to different lifts

4.7-1 StandardValuesBrauerCharacter
4.7-2 Frobenius character values

4.8 Known factorizations of multiplicative group orders

4.8-1 CANFACT

4.9 Some loops for StandardFF

4.9-1 Computing all fields in various ranges

4.10 Undocumented features

References

Index

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