标题： 高效检查分离不定元 显示英文标题

标题： Efficiently Checking Separating Indeterminates

摘要： 在本文中，我们继续发展一种新的通过代入计算消去理想的技巧，该技巧被称为$Z$-分离重新嵌入。 给定一个域 $K$ 上多项式环 $K[x_1,\dots,x_n]$ 中的理想 $I$，此方法搜索具有以下性质的不定元元组 $Z=(z_1,\dots,z_s)$： $I$ 包含形如 $f_i = z_i - h_i$ 的多项式，其中 $i=1,\dots,s$ 使得 $h_i$ 中的任何项都不被 $Z$ 中的不定元整除。 由于通常有许多候选元组$Z$，本文解决的任务是高效地检查给定的元组$Z$是否具有此属性。 我们构建了快速算法，用于检查由$I$的生成元张成的向量空间或稍作扩展的向量空间是否包含所需的多项式$f_i$。 我们还将这些算法扩展到布尔多项式，并将其应用于更快地破译 AES 密码系统的减少轮数版本。 显示英文摘要

摘要： In this paper we continue the development of a new technique for computing elimination ideals by substitution which has been called $Z$-separating re-embeddings. Given an ideal $I$ in the polynomial ring $K[x_1,\dots,x_n]$ over a field $K$, this method searches for tuples $Z=(z_1,\dots,z_s)$ of indeterminates with the property that $I$ contains polynomials of the form $f_i = z_i - h_i$ for $i=1,\dots,s$ such that no term in $h_i$ is divisible by an indeterminate in $Z$. As there are frequently many candidate tuples $Z$, the task addressed by this paper is to efficiently check whether a given tuple $Z$ has this property. We construct fast algorithms which check whether the vector space spanned by the generators of $I$ or a somewhat enlarged vector space contain the desired polynomials $f_i$. We also extend these algorithms to Boolean polynomials and apply them to cryptoanalyse round reduced versions of the AES cryptosystem faster.