Middle-Solving Grobner bases algorithm for cryptanalysis over finite fields

Algebraic cryptanalysis usually requires to recover the secret key by solving polynomial equations. Gröbner bases algorithm is a well-known method to solve this problem. However, a serious drawback exists in the Gröbner bases based algebraic attacks, namely, any information won’t be got if we couldn’t work out the Gröbner bases of the polynomial equations system. In this paper, firstly, a generalized model of Gröbner basis algorithms is presented, which provides us a platform to analyze and solve common problems of the algorithms. Secondly, we give and prove the degree bound of the polynomials appeared during the computation of Gröbner basis after field polynomials is added. Finally, by detecting the temporary basis during the computation of Gröbner bases and then extracting the univariate polynomials contained unique solution in the temporary basis, a heuristic strategy named Middle-Solving is presented to solve these polynomials at each iteration of the algorithm. Farther, two specific application mode of Middle-Solving strategy for the incremental and non-incremental Gröbner bases algorithms are presented respectively. By using the Middle-Solving strategy, even though we couldn’t work out the final Gröbner bases, some information of the variables still leak during the computational process.