Several classes of permutation pentanomials with the form [formula omitted] over [formula omitted

Abstract

In this paper, we study the permutation property of pentanomials with the form xrh(xpm−1) over Fp2m, where p∈{2,3}. More precisely, based on some seventh-degree and eighth-degree irreducible pentanomials over F2, we present eight classes of permutation pentanomials over F22m by determining the solutions of some equations with low degrees. In addition, based on the investigation of algebraic curves associated with fractional polynomials over finite fields, eight classes of permutation pentanomials over F32m are discovered by choosing some seventh-degree irreducible pentanomials over F3. Finally, several classes of permutation pentanomials and heptanomials over F22m and F32m are derived from known permutation polynomials on μ2m+1 and μ3m+1, respectively, where μd is the set of d-th roots of unity.