有限域上密码分析的中间解Grobner基算法
Middle-Solving Grobner bases algorithm for cryptanalysis over finite fields
摘要：
代数密码分析通常需要通过求解多项式方程组来恢复秘密密钥。Grobner基算法是解决这一问题的一种著名方法。然而，基于Grobner基的代数攻击存在一个严重的缺陷，即如果不能求出多项式方程组的Grobner基，就无法得到任何信息。本文首先给出了Grobner基算法的一个广义模型，为分析和解决算法中的常见问题提供了一个平台。其次，我们给出并证明了在Grobner基的计算过程中，在增加域多项式后，所出现的多项式的次数界。最后，通过在Grobner基的计算过程中检测临时基，提取临时基中包含唯一解的一元多项式，提出了一种启发式策略Middle-Solving，在算法的每次迭代中求解这些多项式。在此基础上，分别给出了增量Grobner基算法和非增量Grobner基算法中间求解策略的两种具体应用模式。通过中间求解策略，即使我们不能计算出最终的Grobner基，但在计算过程中仍然会泄漏一些变量的信息。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
