Middle-Solving Grobner bases algorithm for cryptanalysis over finite fields

TL;DR

This paper introduces Middle-Solving, a heuristic strategy that enhances Grobner bases algorithms for algebraic cryptanalysis by extracting useful information during computation, even if the final bases are not fully computed.

Contribution

It presents a generalized model of Grobner basis algorithms, proves degree bounds, and introduces Middle-Solving to improve cryptanalysis effectiveness.

Findings

Middle-Solving extracts variable information during Grobner basis computation.

The strategy applies to both incremental and non-incremental algorithms.

It enables partial information recovery even when final bases are not obtained.

Abstract

Algebraic cryptanalysis usually requires to recover the secret key by solving polynomial equations. Grobner bases algorithm is a well-known method to solve this problem. However, a serious drawback exists in the Grobner bases based algebraic attacks, namely, any information won't be got if we couldn't work out the Grobner bases of the polynomial equations system. In this paper, firstly, a generalized model of Grobner 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 Grobner basis after field polynomials is added. Finally, by detecting the temporary basis during the computation of Grobner 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 Grobner bases algorithms are presented respectively. By using the Middle-Solving strategy, even though we couldn't work out the final Grobner bases, some information of the variables still leak during the computational process.