Many systems of interest in cryptography consist of equations of the same degree. Under the assumption that the degree of regularity is finite, we prove upper bounds on the degree of regularity of a system of equations of the same degree, with or without adding the field equations to the system. The bounds translate into upper bounds on the solving degree of the systems, and hence on the complexity of solving them via Gröbner bases methods. Our bounds depend on the number of equations in the system, the number of variables, and the degree of the equations.

翻译：密码学中许多重要系统由同次方程构成。在正则度有限的假设下，我们证明了同次方程组（无论是否添加域方程）正则度的上界。这些上界可转化为方程组求解次数的上界，进而转化为通过Gröbner基方法求解的复杂度上界。我们的上界取决于方程组中方程的数量、变量的数量以及方程的次数。