We study the last fall degrees of {\em semi-local} polynomial systems, and the computational complexity of solving such systems for closed-point and rational-point solutions, where the systems are defined over a finite field. A semi-local polynomial system specifies an algebraic set which is the image of a global linear transformation of a direct product of local affine algebraic sets. As a special but interesting case, polynomial systems that arise from Weil restriction of algebraic sets in an affine space of low dimension are semi-local. Such systems have received considerable attention due to their application in cryptography. Our main results bound the last fall degree of a semi-local polynomial system in terms of the number of closed point solutions, and yield an efficient algorithm for finding all rational-point solutions when the prime characteristic of the finite field and the number of rational solutions are small. Our results on solving semi-local systems imply an improvement on a previously known polynomial-time attack on the HFE (Hidden Field Equations) cryptosystems. The attacks implied in our results extend to public key encryption functions which are based on semi-local systems where either the number of closed point solutions is small, or the characteristic of the field is small. It remains plausible to construct public key cryptosystems based on semi-local systems over a finite field of large prime characteristic with exponential number of closed point solutions. Such a method is presented in the paper, followed by further cryptanalysis involving the isomorphism of polynomials (IP) problem, as well as a concrete public key encryption scheme which is secure against all the attacks discussed in this paper.

翻译：我们研究半局部多项式系统的最后下降度，以及求解此类系统闭点与有理点解的计算复杂性，其中系统定义在有限域上。半局部多项式系统确定了一个代数集，该代数集是局部仿射代数集直积的全局线性变换的像。作为一个特殊但有趣的情形，由低维仿射空间中代数集的Weil限制导出的多项式系统即为半局部系统。此类系统因其在密码学中的应用而备受关注。我们的主要结果通过闭点解的数量界定了半局部多项式系统的最后下降度，并在有限域的特征素数及有理点解数量较小时，给出一种高效算法以求出所有有理点解。关于求解半局部系统的结果改进了先前已知的针对HFE（隐域方程）密码系统的多项式时间攻击方法。我们结果所隐含的攻击可扩展到基于半局部系统的公钥加密函数，其中闭点解数量较少或域的特征数较小。基于大特征素数有限域上具有指数级闭点解的半局部系统构建公钥密码系统仍是可行的。本文提出此类方法，并进一步涉及多项式同构（IP）问题的密码分析，以及一个能抵御本文所讨论所有攻击的具体公钥加密方案。