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A class of new permutation trinomials. (English) Zbl 1380.05002
Summary: In this paper, we characterize the coefficients of $$f(x)=x+a_1 x^{q(q-1)+1}+a_2 x^{2(q-1)+1}$$ in $$\mathbb{F}_{q^2}[x]$$ for even $$q$$ that lead $$f(x)$$ to be a permutation of $$\mathbb{F}_{q^2}$$. We transform the problem into studying some low-degree equations with variable in the unit circle, which are intensively investigated with some parameterization techniques. From the numerical results, the coefficients that lead $$f(x)$$ to be a permutation appear to be completely characterized in this paper. It is also demonstrated that some permutations proposed in this paper are quasi-multiplicative (QM) inequivalent to the previously known permutation trinomials.

##### MSC:
 11T06 Polynomials over finite fields
##### Keywords:
permutation polynomial; finite field; permutation trinomial
Full Text:
##### References:
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