This project is to be replaced in the more general context of information protection. Its research areas are cryptography and symbolic computation. We are here essentially - but not exclusively - concerned with public key cryptography. Public key cryptography relies on the notion of (trapdoor) one-way function. Intuitively, such a function is easy (polynomial-time) to compute, but difficult (at best sub-exponential) to invert (except if one knows a trapdoor, rendering preimage computation easy). One way functions themselves are constructed from hard problems (roughly speaking, problems for which no polynomial-time algorithm is known). In the cryptographic context, although quite a few problems have been proposed to construct primitives, those effectively used are essentially factorization (e.g. in RSA) and discrete logarithm (e.g. in Diffie-Hellman key-exchange). It is well-known that, although polynomial-time algorithms for those problems have not yet been found, they are not safe from a theoretic breakthrough, that would endanger the security of the corresponding schemes.Besides, the schemes based on those two problems are too slow to be used in strongly constrained environments, like mobile communications (ad-hoc networks, RFID...).Thus, one of the main issues in public key cryptography is to identify hard problems, and propose new schemes that are not based on number theory. Having in mind the numerous applications of cryptography in the real world (access control, authentication, securing Internet transactions, protecting communications...),the consequences of new results in this area are not only of scientific interest, but are also important from an economical point of view. Following this line of research, multivariate schemes have been introduced in the mid eighties by Diffie and Fell in 1985, Matsumoto and Imai in 1988. We recall that multivariate cryptography comprises any cryptographic scheme that uses multivariate polynomial systems. It has to be mentioned that schemes based on the hard problem of solving multivariate equations over a finite field are not concerned with the quantum computer threat, whereas as it is well known that number theoretic-based schemes like RSA, DH, or ECDH are. Besides, multivariate schemes enjoy low computational requirements and can yield short signatures, an everlasting concern in public-key cryptography. Recently, multivariate cryptography has become a very important research area. This is mainly due to the fact that a European project (NESSIE) has advised in 2003 to use such a scheme (namely, SFLASH) in the smart-card context. Moreover, there have been proposals for almost every functionality : signature (Sflash, Quartz, TTS, STS, enTTS, UOV, Rainbow), encryption (Hfe, MI, PMI, 2R...), stream cipher (Quad), traitor tracing (Billet-Gilber}). It has to be noted, though, that the situation is quite different in industry : people are still reluctant to use those schemes, because their security is still not well mastered. In order to evaluate the security of new proposed schemes, strong and efficient cryptanalytic methods have to be developped. The main theme we shall address in this project is the evaluation of the security of cryptographic primitives by means of algebraic methods. The idea is to model a cryptographic primitive as a system of algebraic equations. The system is constructed in such a way as to have a correspondence between the solutions of this system, and a secret information of the considered primitive. Once this modeling is done, the problem is then to solve an algebraic system. Our approach is motivated by the fact that algebraic analysis has appeared only a few years ago in the cryptographic setting, and thus constitutes a new way of investigating the security of cryptographic primitives. Moreover its scope is not well understood, and the results in this area are somewhat fuzzy.What we aim at is primarily identifying those schemes that resist algebraic cryptanalysis. For those, we shall then give suitable parameters for secure use, a crucial point that often lacks clarity in the proposed schemes. But in a broader understanding, our project aims at conceiving the best known attacks on a random system of equations. In a first phase of the project, we shall identify the schemes or classes of schemes that are worth studying, from the point of view of algebraic analysis. Then, we shall start the core of our study, namely the analysis itself. This work will be organized according to the classes we have identified at first stage.Our analytic work will also include a prospective part, in which we shall broaden our study by considering schemes arising from secret key cryptography (block ciphers like AES, stream ciphers like QUAD) or other cryptographic primitives (e.g. SHA1). In a subsequent phase, we shall use our results to design, or at least set up elements for the design of secure multivariate schemes.