# A general construction of permutation polynomials of the form (x^2^m+x+δ)^i(2^m-1)+1+x over _2^2m

Recently, there has been a lot of work on constructions of permutation polynomials of the form (x^2^m+x+δ)^s+x over the finite field _2^2m, 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 constructions of the parameter i, we give a general sufficient condition on i such that (x^2^m+x+δ)^i(2^m-1)+1+x permutes _2^2m, that is, (2^k+1)i ≡ 1 or 2^k (mod 2^m+1), where 1 ≤ k ≤ m-1 is any integer. This generalizes a recent result obtained by Gupta and Sharma who actually dealt with the case k=2. It turns out that most of previous constructions of the parameter i are covered by our result, and it yields many new classes of permutation polynomials as well.

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