Constructions of complete permutation polynomials

Full classification of permutation rational functions and complete rational functions of degree three over finite fields

“…Let q be a prime power and F q be the finite field of order q. Writing permutation polynomial maps of F q (and their generalizations) is a matter of great interest in number theory and applied areas (see for example [5,6,8,12,13,15,17,20,21,23,24]). The classification of permutation polynomials of degree 3 has been longly known (see for example [16, Table 7.1]); the vast literature on the topic contains sparse results that, put together, can be used to describe exceptional functions of certain degrees, for example in terms of Rédei functions.…”

A New Method of Construction of Permutation Trinomials with Coefficients 1

“…For integer s, d > 0 such that sd = q − 1, let µ d = {x ∈ F q : x d = 1} denote the unique cyclic subgroup of F × q of order d. Especially when q = 2 2m for some positive integer m and d = 2 m + 1, µ d is written as U in this paper. Permutation polynomials have wide applications in various areas of mathematics and engineering, such as coding theory [17], cryptography [7,16] and combinatorial designs [2]. Permutation binomials and trinomials with coefficients 1 have the simplest algebraic form.…”

Permutation polynomials and complete permutation polynomials over $\mathbb{F}_{q^{3}}$

“…Finding new permutation polynomials and complete permutation polynomials is of great interest in both theoretical and applied aspects. Many constructions of permutation polynomials appeared in the recent years; see for instance [1,7,15,27,32,34]. The reader may refer to [21,Chapter 7], [24,Chapter 8], [16] and references therein for more information.…”