In the complexity estimation for an attack that reduces a cryptosystem to solving a system of polynomial equations, the degree of regularity and an upper bound of the first fall degree are often used in cryptanalysis. While the degree of regularity can be easily computed using a univariate formal power series under the semi-regularity assumption, determining an upper bound of the first fall degree requires investigating the concrete syzygies of an input system. In this paper, we investigate an upper bound of the first fall degree for a polynomial system over a sufficiently large field. In this case, we prove that the first fall degree of a non-semi-regular system is bounded above by the degree of regularity, and that the first fall degree of a multi-graded polynomial system is bounded above by a certain value determined from a multivariate formal power series. Moreover, we provide a theoretical assumption for computing the first fall degree of a polynomial system over a sufficiently large field.

翻译：在将密码系统攻击简化为求解多项式方程组的复杂度估计中，正则度与首降度上界常被用于密码分析。虽然正则度可在半正则性假设下通过单变量形式幂级数轻松计算，但确定首降度上界需要深入考察输入系统的具体合冲关系。本文研究了充分大域上多项式系统的首降度上界。在此情形下，我们证明了非半正则系统的首降度以正则度为上界，且多重分次多项式系统的首降度以多元形式幂级数确定的特定值为上界。此外，我们提出了计算充分大域上多项式系统首降度的理论假设。