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Algebraic Immunity of S-Boxes Based on Power Mappings: Analysis and Construction

机译：基于功率映射的S-Box代数抗扰度：分析与构建

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摘要

The algebraic immunity of an S-box depends on the number and type of linearly independent multivariate equations it satisfies. In this paper, techniques are developed to find the number of linearly independent, multivariate, bi-affine, and quadratic equations for S-boxes based on power mappings. These techniques can be used to prove the exact number of equations for any class of power mappings. Two algorithms to calculate the number of bi-affine and quadratic equations for any $(n,n)$ S-box based on power mapping are also presented. The time complexity of both algorithms is only $O(n^2)$ . To design algebraically immune S-boxes, four new classes of S-boxes that guarantee zero bi-affine equations and one class of S-boxes that guarantees zero quadratic equations are presented. The algebraic immunity of power mappings based on Kasami, Niho, Dobbertin, Gold, Welch, and inverse exponents are discussed along with other cryptographic properties and several cryptographically strong S-boxes are identified. It is conjectured that a known Kasami-like highly nonlinear power mapping is differentially $4$ -uniform. Finally, an open problem to find an $(n,n)$ bijective nonlinear S-box with more than $5n$ quadratic equations is solved.
机译：S-box的代数免疫性取决于它满足的线性独立多元方程的数量和类型。在本文中，开发了一些技术来根据功率映射找到S盒的线性独立，多元，双仿射和二次方程的数量。这些技术可用于证明任何类别的功率映射的方程式的精确数量。还提出了两种基于功率映射来计算任何$（n，n）$ S盒的双仿射和二次方程数的算法。两种算法的时间复杂度仅为$ O（n ^ 2）$。为了设计代数免疫S盒，提出了四类新的S盒，它们保证零仿射方程组和一类S盒，它们保证了零二次方程组。讨论了基于Kasami，Niho，Dobbertin，Gold，Welch和反指数的幂映射的代数免疫性，以及其他密码学性质，并确定了几个密码学强的S盒。可以推测，已知的类似Kasami的高度非线性功率映射是差分$ 4 $均匀的。最后，解决了一个开放问题，该问题找到一个具有大于$ 5n $二次方程的$（n，n）$双射非线性S-box。

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