Mathematically constructed S-boxes arise from algebraic structures and finite field theory to ensure strong, provable cryptographic properties. These mathematically grounded constructions allow for generation of thousands of S-Boxes with high nonlinearity, APN properties, and balanced avalanche characteristics, unlike fully random methods, which lack such theoretical guarantees in exchange for low complexity and more varied results. In this work, we compare mathematically constructed constructions with randomly generated ones to evaluate the relative weakness of the latter. We also establish an average measure of performance for randomly generated permutations, as well as random with forced cycle constraints, and compare them to well-established designs in a simple SPN setting.

翻译：数学构造的S盒源于代数结构与有限域理论，以确保强大且可证明的密码学特性。与完全随机方法相比，这种基于数学的构造能够生成数千个具有高非线性度、APN特性及平衡雪崩特性的S盒，而完全随机方法虽具有低复杂度和更丰富的结果，却缺乏此类理论保证。本研究通过比较数学构造方法与随机生成方法，评估后者的相对弱点。同时，我们建立了随机生成置换以及带强制循环约束的随机置换的平均性能度量标准，并在简单SPN结构下将其与成熟设计方案进行对比。