Permutation polynomials over finite fields are an interesting subject due to their important applications in the areas of mathematics and engineering. In this paper we investigate the trinomial $f(x)=x^{ (p-1)q+1}+x^{ pq}-x^{ q+(p-1)}$ over the finite field $\mathbb{ F}_{ q^2}$, where $p$ is an odd prime and $q=p^k$ with $k$ being a positive integer. It is shown that when $p=3$ or $5$, $f(x)$ is a permutation trinomial of $\mathbb{ F}_{ q^2}$ if and only if $k$ is even. This property is also true for more general class of polynomials $g(x)=x^{ (q+1)l+(p-1)q+1}+x^{ (q+1)l+pq}-x^{ (q+1)l+q+(p-1)}$, where $l$ is a nonnegative integer and $\gcd(2l+p,q-1)=1$. Moreover, we also show that for $p=5$ the permutation trinomials $f(x)$ proposed here are new in the sense that they are not multiplicative equivalent to previously known ones of similar form.
Categories: cs.IT math.IT