Unicyclic Strong Permutations

TL;DR

This paper introduces a class of permutations over finite fields called unicyclic strong permutations, analyzing their algebraic properties, cycle structure, and empirical behavior for certain dimensions, with implications for cryptographic applications.

Contribution

It defines and studies unicyclic strong permutations, proving their high algebraic degree, average number of algebraic normal form terms, and providing empirical analysis for specific cases.

Findings

Permutations have high algebraic degree of n-1.

Average number of terms approaches 2^{n-1}.

Empirical results show limitations for even n, with successful constructions for odd n up to 11.

Abstract

In this paper, we study some properties of a certain kind of permutation $\sigma$ over $\mathbb{F}_{2}^{n}$, where $n$ is a positive integer. The desired properties for $\sigma$ are: (1) the algebraic degree of each component function is $n-1$; (2) the permutation is unicyclic; (3) the number of terms of the algebraic normal form of each component is at least $2^{n-1}$. We call permutations that satisfy these three properties simultaneously unicyclic strong permutations. We prove that our permutations $\sigma$ always have high algebraic degree and that the average number of terms of each component function tends to $2^{n-1}$. We also give a condition on the cycle structure of $\sigma$. We observe empirically that for $n$ even, our construction does not provide unicylic permutations. For $n$ odd, $n \leq 11$, we conduct an exhaustive search of all $\sigma$ given our construction for specific examples of unicylic strong permutations. We also present some empirical results on the difference tables and linear approximation tables of $\sigma$.