## Section: New Results

### Solving Polynomial Systems over Finite Fields: Improved Analysis of the Hybrid Approach

The Polynomial System Solving (PoSSo) problem is a fundamental
NP-Hard problem in computer algebra. Among others, PoSSo have
applications in area such as coding theory and cryptology.
Typically, the security of multivariate public-key schemes (MPKC)
such as the UOV cryptosystem of Kipnis, Shamir and Patarin is
directly related to the hardness of PoSSo over finite fields. The
goal of [22] is to further understand the influence of
finite fields on the hardness of PoSSo. To this end, we consider
the so-called *hybrid approach*. This is a polynomial system
solving method dedicated to finite fields proposed by Bettale,
Faugère and Perret (Journal of Mathematical Cryptography, 2009).
The idea is to combine exhaustive search with Gröbner bases. The
efficiency of the hybrid approach is related to the choice of a
trade-off between the two methods. We propose here an improved
complexity analysis dedicated to quadratic systems. Whilst the
principle of the hybrid approach is simple, its careful analysis
leads to rather surprising and somehow unexpected results. We prove
that the optimal trade-off (i.e. number of variables to be fixed)
allowing to minimize the complexity is achieved by fixing a number
of variables proportional to the number of variables of the system
considered, denoted $n$. Under some natural algebraic assumption, we
show that the asymptotic complexity of the hybrid approach is
${2}^{(3.31-3.62\phantom{\rule{0.166667em}{0ex}}{log}_{2}{\left(q\right)}^{-1})\phantom{\rule{0.166667em}{0ex}}n}$, where $q$ is the
size of the field (under the condition in particular that
$log\left(q\right)\ll n$). This is to date, the best complexity for solving
PoSSo over finite fields (when $q>2$). We have been able to
quantify the gain provided by the hybrid approach compared to a
direct Gröbner basis method. For quadratic systems, we show
(assuming a natural algebraic assumption) that this gain is
exponential in the number of variables. Asymptotically, the gain is
${2}^{1.49\phantom{\rule{0.166667em}{0ex}}n}$ when both $n$ and $q$ grow to infinity and $log\left(q\right)$.