Permutation polynomials over finite fields — A survey of recent advances

Permutation polynomials over finite fields constitute an active research area in which advances are being made constantly. We survey the contributions made to this area in recent years. Emphasis is placed on significant results and novel methods.     

New classes of permutation binomials and permutation trinomials over finite fields

Permutation polynomials over finite fields play important roles in finite fields theory. They also have wide applications in many areas of science and engineering such as coding theory, cryptography, combinatorial design, communication theory and so on. Permutation binomials and permutation trinomials attract people's interest due to their simple algebraic forms and additional extraordinary properties. In this paper, we find a new result about permutation binomials and construct several new classes of permutation trinomials. Some of them are generalizations of known ones.     

Several classes of complete permutation polynomials over finite fields of even characteristic

In this paper, we find three classes of complete permutation polynomials over finite fields of even characteristic. The first class of quadrinomials is complete in the sense of addition. The second and third classes of binomials and trinomials are complete in multiplication. Moreover, a result related to the complete property in multiplication of a special class of polynomials is also given.     

Permutation binomials of the form xr(xq−1 + a) over Fqe

We present several existence and nonexistence results for permutation binomials of the form $x^r(x^{q-1}+a)$, where $e\ge 2$ and $a\in {\mathbb{F}}_{q^e}^{⁎}$. As a consequence, we obtain a complete characterization of such permutation binomials over ${\mathbb{F}}_{q^2}$, ${\mathbb{F}}_{q^3}$, ${\mathbb{F}}_{q^4}$, ${\mathbb{F}}_{p^5}$, and ${\mathbb{F}}_{p^6}$, where p is an odd prime.     

A class of permutation quadrinomials

In this paper, we investigate the permutation behavior of a class of quadrinomials. Each term of these quadrinomials has a Niho-type exponent, and two sets of coefficient triples making the quadrinomials to be permutations are obtained. We use a substitution to transform the permutation problem into the root distribution problem in the unit circle of certain quadratic and cubic equations.     

Permutation polynomials from trace functions over finite fields

In this paper, we propose several classes of permutation polynomials based on trace functions over finite fields of characteristic 2. The main result of this paper is obtained by determining the number of solutions of certain equations over finite fields.