## Section: Overall Objectives

### Introduction

The main objective of the POLSYS project is to solve systems of polynomial equations. Our main objectives are:

**Fundamental Algorithms and Structured Systems.**The objective is to propose fast exponential exact algorithms to solve the problem of solving polynomial equations and to identify large classes of structured polynomial systems which can be solved in polynomial time.**Solving Systems over the Reals and Applications.**For positive dimensional systems basic questions over the reals may be very difficult (for instance testing the existence of solutions) but also very useful in applications (e.g. global optimization problems). We plan to propose efficient algorithms and implementations to address the most important issues: computing sample points in the real solution sets, decide if two such sample points can be path-connected and, as a long term objective, perform quantifier elimination over the reals (computing a quantifier-free formula which is equivalent to a given quantified boolean formula of polynomial equations/inequalities).**Dedicated Algebraic Computation and Linear Algebra.**While linear algebra is a key step in the computation of Gröbner bases, the matrices generated by the algorithms ${F}_{4}/{F}_{5}$ have specific structures (quasi block triangular).The objective is to develop a dedicated efficient multi-core linear algebra package as the basis of a future open source library for computing Gröbner bases.**Solving Systems in Finite Fields, Applications in Cryptology and Algebraic Number Theory.**We propose to develop a systematic use of*structured systems*in Algebraic Cryptanalysis. We want to improve the efficiency and to predict the theoretical complexity of such attacks. We plan to demonstrate the power of algebraic techniques in new areas of cryptography such as Algebraic Number Theory (typically, in curve based cryptography).