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General constructions of permutation polynomials of the form $$(x^{2^m} + x + \delta)^{i(2^m - 1) + 1} + x$$ over $$\mathbb{F}_{2^{2 m}}$$. (English) Zbl 1388.05009
Summary: Recently, there has been a lot of work on constructions of permutation polynomials of the form $$(x^{2^m} + x + \delta)^s + x$$ over the finite field $$\mathbb{F}_{2^{2 m}}$$, especially in the case when $$s$$ is of the form $$s = i(2^m - 1) + 1$$ (Niho exponent). In this paper, we further investigate permutation polynomials with this form. Instead of seeking for sporadic construction of the parameter $$i$$, we give two general sufficient conditions on $$i$$ such that $$(x^{2^m} + x + \delta)^{i(2^m - 1) + 1} + x$$ permutes $$\mathbb{F}_{2^{2 m}}$$: (i) $$(2^k + 1) i \equiv 1$$ or $$2^k \pmod{2^m + 1}$$; (ii) $$(2^k - 1) i \equiv - 1$$ or $$2^k \pmod{2^m + 1}$$, where $$1 \leq k \leq m - 1$$ is any integer. It turns out that most of previous constructions of the parameter $$i$$ are covered by our results, and they yield many new classes of permutation polynomials as well.

##### MSC:
 05A05 Permutations, words, matrices 11T06 Polynomials over finite fields 11T55 Arithmetic theory of polynomial rings over finite fields
##### Keywords:
finite field; permutation polynomial; Niho exponent
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##### References:
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