A link between two classes of permutation polynomials

Abstract

The study of permutation polynomials over finite fields has attracted many scholars' attentions due to their wide applications and there are several different forms of permutations over finite fields. However, there is little literature on the relation between different forms of permutations.

In this paper, we find an equivalent relation between permutation polynomials of the form $f(x)=xh\left(x^{2^n-1}\right)$ over ${\mathbb{F}}_{2^{2n}}$ and permutations of the form $F(x)=\frac{H_1(\mathrm{\Delta })}{H_1(\mathrm{\Delta })+H_2(\mathrm{\Delta })}+x$ on ${\mathbb{F}}_{2^n}$, where $\mathrm{\Delta }=\frac{1}{x^2+x+\delta }$, $\delta \in {\mathbb{F}}_{2^n}$ with ${\mathrm{Tr}}_{2^n}(\delta )=1$ and $H_1,H_2$ are the x-coordinate and 1-coordinate of h (see Definition 2.4), respectively. A special case with $H_1+H_2=1$, which means $F(x)=H_1\left(\frac{1}{x^2+x+\delta }\right)+x$, has been already studied [8], [16], [21], [22]. We summarize the known permutations and then construct 8 new classes of permutations of such form, from which one can derive some permutation polynomials of the form $xh\left(x^{2^n-1}\right)$ over ${\mathbb{F}}_{2^{2n}}$ directly. Finally, we give an asymptotic result about ${\left(\frac{1}{x^2+x+\delta }\right)}^{2^k}+x$ to be a permutation over ${\mathbb{F}}_{2^n}$ using the Hasse-Weil bound.