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## 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.