In this paper, we review and explain the existing algebraic cryptanalysis of multivariate cryptosystems from the hidden field equation (HFE) family. These cryptanalysis break cryptosystems in the HFE family by solving multivariate systems of equations. In this paper we present a new and efficient attack of this cryptosystem based on fast algorithms for computing Gröbner basis. In particular it was was possible to break the first HFE challenge (80 bits) in only two days of CPU time by using the new algorithm F5 implemented in C. From a theoretical point of view we study the algebraic properties of the equations produced by instance of the HFE cryptosystems and show why they yield systems of equations easier to solve than random systems of quadratic equations of the same sizes. Moreover we are able to bound the maximal degree occuring in the Gröbner basis computation. As a consequence, we gain a deeper understanding of the algebraic cryptanalysis against these cryptosystems. We use this understanding to devise a specific algorithm based on sparse linear algebra. In general, we conclude that the cryptanalysis of HFE can be performed in polynomial time. We also revisit the security estimates for existing schemes in the HFE family.

This paper introduces a new efficient algorithm for computing Gröbner bases. To avoid as much intermediate computation as possible, the algorithm computes successive truncated Gröbner bases and it replaces the classical polynomial reduction found in the Buchberger algorithm by the simultaneous reduction of several polynomials. This powerful reduction mechanism is achieved by means of a symbolic precomputation and by extensive use of sparse linear algebra methods. Current techniques in linear algebra used in Computer Algebra are reviewed together with other methods coming from the numerical field. Some previously untractable problems (Cyclic 9) are presented as well as an empirical comparison of a first implementation of this algorithm with other well known programs. This comparison pays careful attention to methodology issues. All the benchmarks and CPU times used in this paper are frequently updated and available on a Web page. Even though the new algorithm does not improve the worst case complexity it is several times faster than previous implementations both for integers and modulo p computations.

We present a new proof of the algebraic part of a symmetry theorem for the central configurations of the newtonian planar 4-body problem with equal masses, using Grobner bases. This approach is used to obtain a new symmetry theorem for the central configurations of the newtonian spatial 5-body problem with equal masses in the convex case. In fact we prove a more general statement of the theorem, valid for a class of potentials defined by functions with increasing and concave derivatives.

The design of two-dimensional (2-D) filter banks yielding orthogonality and linear-phase filters and generating regular wavelet bases is a difficult task involving algebraic properties of multivariate polynomials, Using cascade forms implies dealing with nonlinear optimization, We turn the issue of optimizing the orthogonal linear-phase cascade from Kovacevic and Vetterli into a polynomial problem and solve it using Grobner basis techniques and computer algebra, This leads to a complete description of maximally hat wavelets among the orthogonal linear-phase family proposed by Kovacevic and Vetterli. We obtain up to five degrees of flatness for a 16 x 16 filter bank, whose Sobolev exponent is 2.11, making this wavelet the most regular orthogonal linear-phase nonseparable wavelet to the authors' knowledge.

The classes of parallel manipulators are listed, which are defined by the constraint that some legs have the same fixation on the platform or on the base. For each class, the generic number of assembly configurations is given. It appears that this number never changes if the constraint of planarity is added.Grobner base computations are widely used for obtaining these results. It appears that this method is now an efficient tool for solving polynomial systems which appear in robot kinematics.

We present an efficient algorithm for the transformation of a Gröbner basis of a zero-dimensional ideal with respect to any given ordering into a Gröbner basis with respect to any other ordering. This algorithm is polynomial in the degree of the ideal. In particular the lexicographical Gröbner basis can be obtained by applying this algorithm after a total degree Gröbner basis computation: it is usually much faster to compute the basis this way than with a direct application of Buchberger's algorithm.