Algebraic cryptanalysis of HFE using Gröbner bases
Abstract
HFE (Hidden Fields Equations) is a public key cryptosystem using (multivariat- e) polynomial operations over finite fields. It has been proposed by Jacques Patarin following the ideas of Matsumoto and Imai. In this paper we present a new and efficient attack of this cryptosystem based on fast algorithms for computing Gröbner basis. The attack consists simply in computing a Gröbner basis of the public key. Of course the efficiency of this attack depends strongly on the choice of the algorithm for computing the Gröbner basis: while the corresponding algebraic systems are completely far beyond the capacity of any implementation of the Buchberger algorithm, 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. We establish experimentally that the algebraic systems coming from HFE behave not as «random systems» so that they can be solved in polynomial time when the degree d of the univariate polynomial is fixed. For practical value of d we can establish precisely the complexity of this attack: O(n^8) (resp. O(n^10)) when 16