More classes of permutation pentanomials over finite fields with characteristic two

Abstract

Let $q=2^m$. In this paper, we investigate permutation pentanomials over $F_{q^2}$ of the form $f\left(x\right)=x^t+x^{r_1(q-1)+t}+x^{r_2(q-1)+t}+x^{r_3(q-1)+t}+x^{r_4(q-1)+t}$ with $\gcd (x^{r_4}+x^{r_3}+x^{r_2}+x^{r_1}+1,x^t+x^{t-r_1}+x^{t-r_2}+x^{t-r_3}+x^{t-r_4})=1$. We transform the problem concerning permutation property of $f\left(x\right)$ into demonstrating that the corresponding fractional polynomial permutes the unit circle U of $F_{q^2}$ with order $q+1$ via a well-known lemma, and then into showing that there are no certain solution in $F_q$ for some high-degree equations over $F_q$ associated with the fractional polynomial. According to numerical data, we have found all such permutations with $4\le t<100,1\le r_i\le t$, $i\in [1,4]$. Several permutation polynomials are also investigated from the fractional polynomials permuting the unit circle U found in this paper.