代数密码分析中的形式幂级数
Formal Power Series on Algebraic Cryptanalysis
AI总结：
本文研究了多项式系统在足够大域上的第一个跌落度的上界，通过多变量形式幂级数确定其上限，并提出了计算该度数的理论假设。
摘要：
在估计将密码系统还原为求解多项式方程组的攻击复杂度时，通常使用正则度和第一个跌落度的上界。虽然在半正则性假设下，可以通过单变量形式幂级数轻松计算正则度，但确定第一个跌落度的上界需要研究输入系统的具体syzygies。在本文中，我们研究了多项式系统在足够大的域上的第一个跌落度的上界。在这种情况下，我们证明了非半正则系统的第一个跌落度上限由正则度决定，而多格雷度多项式系统的第一个跌落度上限则由多变量形式幂级数确定的某个值决定。此外，我们为计算足够大的域上多项式系统的第一个跌落度提供了理论假设。
英文摘要：
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.
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