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 $s, t$

On Equivalence of Known Families of APN Functions in Small Dimensions

In this extended abstract, we computationally check and list the CCZ-inequivalent APN functions from infinite families on $\mathbb{F}_2^n$ for n from 6 to 11. These functions are selected with simplest coefficients from CCZ-inequivalent classes. This work can simplify checking CCZ-equivalence between any APN function and infinite APN families.Comment: This paper is already in "PROCEEDING OF THE 20TH CONFERENCE OF FRUCT ASSOCIATION