A multivariate cryptograpic instance in practice is a multivariate polynomial system. So the security of a protocol rely on the complexity of solving a multivariate polynomial system. In this paper there is an overview on a general algorithm used to solve a multivariate system and the quantity to which the complexity of this algorithm depends on: the solving degree. Unfortunately, it is hard to compute. For this reason, it is introduced an invariant: the degree of regularity. This invariant, under certain condition, give us an upper bound on the solving degree. Then we speak about random polynomial systems and in particular what "random" means to us. Finally, we give an upper bound on both the degree of regularity and the solving degree of such random systems.

翻译：实践中，多元密码学实例通常表现为多元多项式系统。因此，协议的安全性取决于求解多元多项式系统的复杂度。本文综述了求解多元多项式系统的通用算法，以及该算法复杂度所依赖的关键量：求解次数。然而，求解次数难以精确计算。为此，我们引入一个不变量：正则次数。该不变量在特定条件下可为求解次数提供上界。随后，我们探讨随机多项式系统，特别是“随机”一词在此语境中的具体含义。最后，我们给出了此类随机系统的正则次数与求解次数的上界估计。