Infinite classes of generalised complete permutations

Abstract

In Pasalic et al. (2016) a construction allowing for high levels of modification was presented. It can be used to construct several important combinatorial structures, among them are examples of complete permutations. Here the method is used to construct infinite classes of generalised complete permutations $F(x)$, where both $F(x)$ and $F(x)+h_D(x)$ are permutations, not just $F(x)+x$. In the article three families of the function $h_D(x)$ are considered $h_D(x)$ being a function multiplying the vector $x$ with a vector $D$, a permutation matrix $D$, or a linear mapping matrix $D$. Most existing results related to complete permutations use the finite field notation, while in this article we are developing permutations based on the vector space structure. The case where $D$ is a binary vector needs to be emphasised. In this case the permutation $F(x)$ is defined in such a way that $F(x)+D^{T}x$ remains a permutation for any of the $2n-3$ vectors $D$ as defined in Eq. (5). Let $S$ be the set of all such vectors. We present for arbitrary $n>4$ construction of an $S$-complete permutation $F(x)$, where $|S|=2n-3$. Additionally, we prove that each of these permutations $F$ has such a corresponding linear subspace $L$ that the pair $(F,L)$ satisfies the ($C$)-property and can be used to construct a huge infinite class of bent functions in Carlet’s $\mathcal{C}$ class. In Mandal et al. (2016) it was proven that finding such pairs $(F,L)$ is a difficult problem.

dr. Nastja Cepak

Assistant Professor