Arithmetic of Finite Fields: First International Workshop, WAIFI 2007, Madrid, Spain, June 21-22, 2007. Proceedings

Robert W. Fitzgerald, Joseph L. Yucas (auth.), Claude Carlet, Berk Sunar (eds.)3540730737, 9783540730736

Explicit factorizations, into a product of irreducible polynomials, over Fq of thecyclotomic polynomials Q2n(x) are given in [4] when q ≡ 1 (mod 4). The caseq ≡ 3 (mod 4) is done in [5]. Here we give factorizations of Q2nr(x) where ris prime and q ≡ ±1 (mod r). In particular, this covers Q2n3(x) for all Fq ofcharacteristic not 2, 3. We apply this to get explicit factorizations of the firstand second kind Dickson polynomials of order 2n3 and 2n3 − 1 respectively.Explicit factorizations of certain Dickson polynomials have been used to computeBrewer sums [1]. But our basic motivation is curiosity, to see what factorsarise. Of interest then is how the generalized Dickson polynomials Dn(x, b) arisein the factors of the cyclotomic polynomials and how the Dickson polynomialsof the first kind appear in the factors of both kinds of Dickson polynomials.

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Table of contents :
Front Matter….Pages –
Explicit Factorizations of Cyclotomic and Dickson Polynomials over Finite Fields….Pages 1-10
Some Notes on d -Form Functions with Difference-Balanced Property….Pages 11-17
A Note on Modular Forms on Finite Upper Half Planes….Pages 18-24
A Coprocessor for the Final Exponentiation of the η T Pairing in Characteristic Three….Pages 25-39
VLSI Implementation of a Functional Unit to Accelerate ECC and AES on 32-Bit Processors….Pages 40-54
Efficient Multiplication Using Type 2 Optimal Normal Bases….Pages 55-68
Effects of Optimizations for Software Implementations of Small Binary Field Arithmetic….Pages 69-84
Software Implementation of Arithmetic in ….Pages 85-102
Complexity Reduction of Constant Matrix Computations over the Binary Field….Pages 103-115
Towards Optimal Toom-Cook Multiplication for Univariate and Multivariate Polynomials in Characteristic 2 and 0….Pages 116-133
A Construction of Differentially 4-Uniform Functions from Commutative Semifields of Characteristic 2….Pages 134-146
Complete Mapping Polynomials over Finite Field F 16 ….Pages 147-158
On the Classification of 4 Bit S-Boxes….Pages 159-176
The Simplest Method for Constructing APN Polynomials EA-Inequivalent to Power Functions….Pages 177-188
New Point Addition Formulae for ECC Applications….Pages 189-201
Explicit Formulas for Real Hyperelliptic Curves of Genus 2 in Affine Representation….Pages 202-218
The Quadratic Extension Extractor for (Hyper)Elliptic Curves in Odd Characteristic….Pages 219-236
On Kabatianskii-Krouk-Smeets Signatures….Pages 237-251
Self-certified Signatures Based on Discrete Logarithms….Pages 252-263
Attacking the Filter Generator over GF (2 m )….Pages 264-275
Cyclic Additive and Quantum Stabilizer Codes….Pages 276-283
Determining the Number of One-Weight Cyclic Codes When Length and Dimension Are Given….Pages 284-293
Error Correcting Codes from Quasi-Hadamard Matrices….Pages 294-302
Fast Computations of Gröbner Bases and Blind Recognitions of Convolutional Codes….Pages 303-317
A Twin for Euler’s φ Function in $mathbb{F}_2[X]$ ….Pages 318-332
Discrete Phase-Space Structures and Mutually Unbiased Bases….Pages 333-345
Some Novel Results of p -Adic Component of Primitive Sequences over Z/(p d ) ….Pages 346-353
Back Matter….Pages –