## Section: Scientific Foundations

### Solving Systems in Finite Fields, Applications in Cryptology and Algebraic Number Theory.

Participants : Jean-Charles Faugère, Ludovic Perret, Guénaël Renault, Louise Huot, Frédéric de Portzamparc, Rina Zeitoun.

Here, we focus on solving polynomial systems over finite fields
(i.e. the discrete case) and the corresponding applications
(Cryptology, Error Correcting Codes, ...). Obviously this
objective can be seen as an application of the results of the two
previous objectives. However, we would like to emphasize that it is
also the source of new theoretical problems and practical challenges.
We propose to develop a systematic use of *structured systems* in
*algebraic cryptanalysis*.

*(i)* So far, breaking a cryptosystem using algebraic
techniques could be summarized as modeling the problem by algebraic
equations and then computing a, usually, time consuming Gröbner
basis. A new trend in this field is to require a theoretical
complexity analysis. This is needed to explain the behavior of the
attack but also to help the designers of new cryptosystems to propose
actual secure parameters.

*(ii)* To assess the security of
several cryptosystems in symmetric cryptography (block ciphers, hash
functions, ...), a major difficulty is the size of the systems
involved for this type of attack. More specifically, the bottleneck
is the size of the linear algebra problems generated during a Gröbner basis
computation.

We propose to develop a
systematic use of *structured systems* in *algebraic
cryptanalysis*.

The first objective is to build on the recent breakthrough in attacking McEliece's cryptosystem: it is the first structural weakness observed on one of the oldest public key cryptosystem. We plan to develop a well founded framework for assessing the security of public key cryptosystems based on coding theory from the algebraic cryptanalysis point of view. The answer to this issue is strongly related to the complexity of solving bihomogeneous systems (of bidegree $(1,d)$). We also plan to use the recently gained understanding on the complexity of structured systems in other areas of cryptography. For instance, the MinRank problem – which can be modeled as an overdetermined system of bilinear equations – is at the heart of the structural attack proposed by Kipnis and Shamir against HFE (one of the most well known multivariate public cryptosystem). The same family of structured systems arises in the algebraic cryptanalysis of the Discrete Logarithmic Problem (DLP) over curves (defined over some finite fields). More precisely, some bilinear systems appear in the polynomial modeling the points decomposition problem. Moreover, in this context, a natural group action can also be used during the resolution of the considered polynomial system.

Dedicated tools for linear algebra problems generated during the Gröbner basis computation will be used in algebraic cryptanalysis. The promise of considerable algebraic computing power beyond the capability of any standard computer algebra system will enable us to attack various cryptosystems or at least to propose accurate secure parameters for several important cryptosystems. Dedicated linear tools are thus needed to tackle these problems. From a theoretical perspective, we plan to further improve the theoretical complexity of the hybrid method and to investigate the problem of solving polynomial systems with noise, i.e. some equations of the system are incorrect. The hybrid method is a specific method for solving polynomial systems over finite fields. The idea is to mix exhaustive search and Gröbner basis computation to take advantage of the over-determinacy of the resulting systems.

Polynomial system with noise is currently emerging as a problem of major interest in cryptography. This problem is a key to further develop new applications of algebraic techniques; typically in side-channel and statistical attacks. We also emphasize that recently a connection has been established between several classical lattice problems (such as the Shortest Vector Problem), polynomial system solving and polynomial systems with noise. The main issue is that there is no sound algorithmic and theoretical framework for solving polynomial systems with noise. The development of such framework is a long-term objective.