Unicyclic Strong Permutations
This work addresses a theoretical problem in cryptography and combinatorics for researchers studying permutation properties, but it is incremental as it builds on known constructions with specific constraints.
The paper tackles the problem of constructing permutations over binary vectors with specific algebraic and combinatorial properties, termed unicyclic strong permutations, and proves that their construction yields high algebraic degree and average term counts approaching 2^{n-1}, with empirical searches conducted for odd n up to 11.
In this paper, we study some properties of a certain kind of permutation $σ$ over $\mathbb{F}_{2}^{n}$, where $n$ is a positive integer. The desired properties for $σ$ 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 $σ$ 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 $σ$. 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 $σ$ 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 $σ$.