Algebraic Attacks from a Groebner Basis Perspective

In this paper we propose a new algorithm for computing Groebner basis for a system of multivariate polynomial equations describing a cryptosystem. The objective for designing this algorithm is to reduce the degree and number of polynomials resulting in a Groebner basis, which appears in the output of the algorithm. To attain this goal, a new division algorithm is proposed. The proposed algorithm, improved Buchberger and F4 algorithm have been applied to the system of algebraic equations extracted from the Courtois Toy Cipher and their efficiencies have been compared. The results show that the proposed algorithm has advantages over improved Buchberger and F4 algorithms from the view point of the number of polynomials within the obtained Groebner basis and computational (time) complexity.