Several classes of permutation pentanomials

Abstract

For each prime $p$ and each power $q=p^k$, we present two large classes of permutation polynomials over $\F_{q^2}$ of the form $X^{r}B(X^{q-1})$ which have at most five terms, where $B(X)$ is a polynomial with coefficients in the prime field of $\F_{q^2}$ except at most one.