Arithmetic of Finite Fields

Name: Arithmetic of Finite Fields | 4th International Workshop, WAIFI 2012, Bochum, Germany, July 16-19, 2012, Proceedings
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4th International Workshop, WAIFI 2012, Bochum, Germany, July 16-19, 2012, Proceedings

Ferruh Özbudak Francisco Rodriguez-Henriquez(Herausgeber*in)

Springer (Verlag)

Erschienen am 2. Juli 2012

XII, 247 Seiten

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Inhalt

Title
Preface
Table of Contents
Generalised Jacobians in Cryptography and Coding Theory
Introduction and Summary
Preliminaries
Pairings on Generalised Jacobians
A Generalised Tate-Lichtenbaum Pairing
Variations - New Bilinear and Multilinear Pairings
Weak Discrete Logarithm Problem in the Domain
Lower Complexity Bounds for Codes
Generalised Algebraic Geometric Codes
Reduction of the Discrete Logarithm Problem to Code Based Computational Problems
Efficient Generating Sets of Generalised Jacobians
Examples
References
The Weight Distribution of a Family of Reducible Cyclic Codes
Introduction
Definitions, Notations, Preliminaries and Main Assumption
Some General Results
The Weight Distribution of a Family of Non-irreducible Cyclic Codes
Conclusion
References
A New Method for Constructing Small-Bias Spaces from Hermitian Codes
Introduction
The AG-Bound
The New Small-Bias Spaces
Time Complexity Considerations
Small-Bias Spaces from Norm-Trace Codes
References
An Improved Threshold Ring Signature Schem eBased on Error Correcting Codes
Introduction
Preliminaries
Definitions
The q-SD Identification Scheme
Description of the q-SD Identification Scheme
Signature Schemes from Identification Schemes
Code-Based Threshold Ring Signature Schemes
Description of Our Threshold Identification Protocol
Security
Performance and Implementation
Practical Results
Conclusion
References
Sequences and Functions Derived from Projective Planes and Their Difference Sets
Introduction
Difference Sets and Incidence Structures
Equivalence of Difference Sets and Isomorphism of Incidence Structures
Projections of Generalized Difference Sets and a Lifting Problem
The Prime Power Conjecture for Projective Planes
Difference Sets and Affine Difference Sets
Difference Sets and Semifields
The n Even Case
The n Odd Case
Direct Product Difference Sets
Generalized Difference Sets Relative to Three Subgroups
Conclusion
References
On Formally Self-dual Boolean Functions in 2,4 and 6 Variables
Introduction
Notation and Definitions
Classification of Formally Self-dual Boolean Functions
Conclusion
References
On the Algebraic Normal Form and Walsh Spectrum of Symmetric Functions over Finite Rings
Introduction
Symmetric Functions over Finite Rings
Functions over Finite Rings
Partitions
Symmetric Functions over Finite Rings
Algebraic Normal Form
Definition
Computing the Simplified Value Vector of m
Link between Simplified ANF and Simplified Value Vector
Walsh Spectrum
Definition
Computing the Matrix Wq,r,n
Computing Wr,r,n
Computing the Walsh Spectrum of a Symmetric Function
Using These Algorithms to Test Large Set of Symmetric Functions
Conclusion
References
Verification of Restricted EA-Equivalence for Vectorial Boolean Functions
Introduction
Preliminaries
Verification of Restricted EA-Equivalence
Conclusions
References
Software Implementation of Modular Exponentiation, Using Advanced Vector Instructions Architectures
Introduction
Preliminaries
The RSA Context
The Non Reduced Montgomery Multiplication
The Relevant AVX2 Instructions
Modular Exponentiation with Vector Instructions
Redundant Representation
NRMM
Modular Exponentiation Using VNRMM
Implementation, Choice of Parameters and Optimizations
Choice of Parameters
Why Is the AVX2 Architecture Sufficient for an Efficient Vectorized Implementation?
Optimizing the Implementation
Vectorized Redundant Montgomery Square
Results
Conclusion
References
Efficient Multiplication over Extension Fields
Introduction
AMNS
Definition of an AMNS Representation
Arithmetic in AMNS Representation
Efficient Multiplication in Fqm Using AMNS
Theory
Some Theory about Lattices
Construction When We Can Choose q
Construction When We Cannot Choose q
Implementation and Results
Conclusion
References
GF(2m) Finite-Field Multipliers with Reduced Activity Variations
Introduction
Activity in Hardware Arithmetic Operators
GF(2m) Finite-Field Multiplication Operators
Classic Two-Step Multiplication Algorithm
Montgomery Multiplication Multiplier
Mastrovito Multiplication Algorithm
Useful Activity Analysis for Multiplication Algorithms
Modifications on Multiplication Algorithms for Reducing Useful Activity Variations
Conclusion
References
Finding Optimal Formulae for Bilinear Maps
Introduction
Some Instances of the Bilinear Rank Problem
From Bilinear Applications to Linear Algebra
Solving the Linear Algebra Problem
Naive Algorithm
Improved Algorithm
Implementation Issues
From Solution Subspaces back to Formulae
Complexity Analysis
Special Case of Symmetric Bilinear Maps
Experimental Results
Conclusion
References
Solving Binary Linear Equation Systems over the Rationals and Binaries
Introduction
Background
Deriving a Solution over F2
Discussion
Conclusion
References
Hashing with Elliptic Curve L-Functions
Introduction
Elliptic Curve L-Function
ECOH
Hashing with L-Functions
One Way Function
Hashing and MAC's Protocol
MAC's Elliptic Curve L-Function against ECOH
Implementations:
Security
Conclusion
References
Square Root Algorithms for the Number Field Sieve
The Direct (lifting) Approach
Working p-Adically
Bound on the Square Root Coefficients
Complexity
Couveignes' Algorithm
Montgomery's Algorithm
Iterative Reduction
Complexity
A New CRT-Based Lifting Approach
CRT-Based Reconstruction
Determining Signs
Strategies for Fast Computation
Complexity
Implementation and Experimental Data
A Variant Using a Large Number of Primes
References
Improving the Berlekamp Algorithm for Binomials xn - a
Introduction
Binomials
Berlekamp Algorithm
Our Algorithm
Complexity
Conclusion and Future Works
References
On Some Permutation Binomials of the Form x2n-1k +1 + ax over F2n : Existence and Count
Introduction
Existence of Permutation Binomial x2n-13+1 + ax
Permutation Polynomials of the Form x2n + 2+ax over F22n
Cryptographic Relevance of Permutation Polynomials
Boolean Functions from Permutation Polynomials
Conclusions
References
Author Index

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