## Section: Overall Objectives

### Highlights of the Year

In [4] , we obtain an algorithm to solve Boolean systems with an expected complexity of $O\left({2}^{0.792\phantom{\rule{0.166667em}{0ex}}n}\right)$ breaking the ${2}^{n}$ barrier.

In [10] , we propose an algorithm to solve a variant of the Quantififer Elimination Problem for which the output formula is

*almost equivalent*to the input formula. The complexity of this algorithm is much better than other algorithms and can solve previously untractable problems.In [25] , we improve the complexity of Index Calculus Algorithms in Elliptic Curves by means of Gröbner basis techniques and we analyze the complexity of this new approach by using the multi-homogeneous structure of the equations.