Abstract
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 p q − 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 a more general class of polynomials g(x) = x (q+1)l+(p−1)q+1 + x (q+1)l + p q − 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.
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