Constructions of complete permutations in multiplication

Abstract

Complete permutations in addition over finite fields have attracted many scholars’ attention due to their wide applications in combinatorics, cryptography, sequences, and so on. In 2020, Tu et al. introduced the concept of the complete permutation in the sense of multiplication (CPM for short). In this paper, we further study the constructions and applications of CPMs. We mainly construct many classes of CPMs through three different approaches, i.e., index, self-inverse binomial, which is a new concept proposed in this paper, and linearized polynomial. Particularly, we provide a modular algorithm to produce all CPMs with a given index and determine all CPMs with index 3. Many infinite classes of complete self-inverse binomials are proposed, which explain most of the experimental results about complete self-inverse binomials over \({\mathbb {F}}_{2^n}\) with \(n\le 10\). Six classes of linearized CPMs are given by using standard arguments from fast symbolic computations and a general method is proposed by the AGW criterion. Finally, two applications of CPMs in cryptography are discussed.