一类平移不变置换及其代数结构
A Class of Shift-Invariant Permutations and Their Algebraic Structure
- Hubei University（湖北大学）
- Key Laboratory of Applied Mathematics, Faculty of Mathematics and Statistics, Hubei University（湖北大学数学与统计学院应用数学重点实验室）
- Key Laboratory of Intelligent Sensing System and Security (Hubei University), Ministry of Education（教育部智能感知系统与安全防护重点实验室（湖北大学））
- Institute of Information Engineering, Chinese Academy of Sciences（中国科学院信息工程研究所）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
本文研究由递归映射序列生成的平移不变置换，给出其多项式复合性质的充要条件，建立与商环单位群的同构，并将置换性质归约为多项式计算，统一了百余种构造。
中文摘要 AI 辅助
在对称密码学中，$\mathbb F_2^n$ 上的平移不变置换因其正则性和实现效率而具有吸引力。构造具有显式代数结构的此类置换仍然是一个具有挑战性的问题。本文研究由与景观相关联的递归定义映射序列 $\{\gamma_j\}_{j\ge0}$ 生成的一族 $\mathbb F_2^n$ 上的平移不变变换。首先，我们证明，在关于 $n$ 和景观支撑集的显式维度条件下，序列 $\{\gamma_j\}_{j\ge0}$ 具有多项式复合性质当且仅当对应的具有非零特殊指标的景观在 $\mathbb F_2^n$ 上是拟守恒的。然后，我们确定 $\{\gamma_j\}_{j\ge0}$ 的最终零或周期行为，包括第一个零或周期指标以及最小最终周期，并建立直到第一个关系的线性无关性。当多项式复合性质成立时，我们在由这些映射生成的幺半群中的置换元素群与商环 $\mathbb F_2[z]/\langle p(z)\rangle$ 的单位群之间建立显式同构，其中 $p(z)$ 是由景观确定的单项式或二项式。该框架将置换性质、复合逆、阶和这些映射的迭代归约为多项式计算。最后，我们将结果应用于商环中由二项式和三项式表示的映射，获得显式判据和公式。我们列出了超过一百个具有代表性的平移不变置换构造，恢复并统一了几个先前研究的族。
英文摘要
Shift-invariant permutations of $\mathbb F_2^n$ are attractive in symmetric cryptography because of their regularity and implementation efficiency. Constructing such permutations with an explicit algebraic structure remains a challenging problem. In this paper, we study a family of shift-invariant transformations of $\mathbb F_2^n$ generated by a recursively defined sequence of mappings $\{γ_j\}_{j\ge0}$ associated with landscapes. First, we prove that, under an explicit dimension condition on $n$ and the support of the landscape, the sequence $\{γ_j\}_{j\ge0}$ has the polynomial composition property if and only if the corresponding landscape with nonzero special index is quasi-conserved on $\mathbb F_2^n$. Then, we determine the eventual zero or periodic behavior of $\{γ_j\}_{j\ge0}$, including the first zero or periodic index and the least eventual period, and establish linear independence up to the first relation. When the polynomial composition property holds, we establish an explicit isomorphism between the group of permutation elements in the monoid generated by these mappings and the unit group of a quotient ring $\mathbb F_2[z]/\langle p(z)\rangle$, where $p(z)$ is a monomial or a binomial determined by the landscape. This framework reduces the permutation property, compositional inverse, order, and iterates of these mappings to polynomial computations. Finally, we apply the results to mappings represented by binomials and trinomials in the quotient ring, obtaining explicit criteria and formulas. We tabulate more than one hundred representative constructions of shift-invariant permutations, recovering and unifying several previously studied families.
