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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
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