POLSYS - 2012 - Annual activity report

Project-Team Polsys

Members

Overall Objectives

Introduction
Highlights of the Year

Scientific Foundations

Introduction
Fundamental Algorithms and Structured Systems
Solving Systems over the Reals and Applications.
Low level implementation and Dedicated Algebraic Computation and Linear Algebra.
Solving Systems in Finite Fields, Applications in Cryptology and Algebraic Number Theory.

Application Domains

Cryptology
Engineering sciences

Software

FGb
RAGlib
Epsilon

New Results

The complexity of solving quadratic boolean systems is better than exhaustive search
Improving the Complexity of Index Calculus Algorithms in Elliptic Curves over Binary Fields
On the relation between the MXL family of algorithms and Gröbner basis algorithms
On the Complexity of the BKW Algorithm on LWE
On the Complexity of the Arora-Ge algorithm against LWE
On enumeration of polynomial equivalence classes and their application to MPKC
Cryptanalysis of HFE, Multi-HFE and Variants for Odd and Even Characteristic
Solving Polynomial Systems over Finite Fields: Improved Analysis of the Hybrid Approach
Efficient Arithmetic in Successive Algebraic Extension Fields Using Symmetries
Algebraic Crypanalysis with Side Channels Information
Worst case complexity of the Continued Fraction (CF) algorithm.
Local Generic Position for Root Isolation of Zero-dimensional Triangular Polynomial Systems.
Univariate Real Root Isolation in Multiple Extension Fields
Mixed volume and distance geometry techniques for counting Euclidean embeddings of rigid graphs.
Variant Quantifier Elimination
Global optimization
Gröbner bases and critical points

Bilateral Contracts and Grants with Industry

Oberthur Technologies
Gemalto

Partnerships and Cooperations

National Initiatives
European Initiatives
International Initiatives
International Research Visitors

Dissemination

Scientific Animation
Teaching - Supervision - Juries
Popularization

Bibliography

Publications of the year

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Section: New Results

Efficient Arithmetic in Successive Algebraic Extension Fields Using Symmetries

In [15] we present new results for efficient arithmetic operations in a number field $𝕂$ represented by successive extensions. These results are based on multi-modular and evaluation–interpolation techniques. We show how to use intrinsic symmetries in order to increase the efficiency of these techniques. Applications to splitting fields of univariate polynomials are presented.

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