A recent survey of permutation trinomials over finite fields

Constructing permutation polynomials is a hot topic in the area of finite fields, and permutation polynomials have many applications in different areas. Recently, several classes of permutation trinomials were constructed. In 2015, Hou surveyed the achievements of permutation polynomials and novel methods. But, very few were known at that time. Recently, many permutation binomials and trinomials have been constructed. Here we survey the significant contribution made to the construction of permutation trinomials over finite fields in recent years. Emphasis is placed on significant results and novel methods. The covered material is split into three aspects: the existence of permutation trinomials of the respective forms $x^{r}h(x^{s})$, $\lambda_{1}x^{a}+\lambda_{2}x^{b}+\lambda_{3}x^{c}$ and $x+x^{s(q^{m}-1)+1} +x^{t(q^{m}-1)+1}$, with Niho-type exponents $s, t$

Determination of a Type of Permutation Trinomials over Finite Fields

Let $f=a{\tt x} +b{\tt x}^q+{\tt x}^{2q-1}\in\Bbb F_q[{\tt x}]$. We find explicit conditions on $a$ and $b$ that are necessary and sufficient for $f$ to be a permutation polynomial of $\Bbb F_{q^2}$. This result allows us to solve a related problem. Let $g_{n,q}\in\Bbb F_p[{\tt x}]$ ($n\ge 0$, $p=\text{char}\,\Bbb F_q$) be the polynomial defined by the functional equation $\sum_{c\in\Bbb F_q}({\tt x}+c)^n=g_{n,q}({\tt x}^q-{\tt x})$. We determine all $n$ of the form $n=q^\alpha-q^\beta-1$, $\alpha>\beta\ge 0$, for which $g_{n,q}$ is a permutation polynomial of $\Bbb F_{q^2}$.Comment: 28 page