(The arXiv version is available online.)

A polynomial f(x) over the finite field F_q is called a permutation polynomial is the map u→f(u) induces a permutation on F_q. In a recent paper, J. Marcos presented several families of permutation polynomials. We generalize his constructions and give simpler proofs.

In more detail, Marcos's first two constructions were variants of constructions in my previous paper. We give a common generalization of all these constructions by giving simple criteria which are necessary and sufficient for a polynomial of the form  xⁱ(bx^k(q-1)/d+g(x^(q-1)/d))  to permute F_q. This is based on the observation that polynomials of this form induce mappings on the set of cosets of F_q modulo the multiplicative subgroup of d-th powers. By using instead an additive subgroup of F_q, we obtain a criterion for a polynomial of the form  A(x)+g(B(x))  to permute F_q, where A and B are additive (linearized) polynomials. In case  B=x^q/p+x^q/p2+...+x^p+x  is the trace polynomial from F_q to its prime field F_p, and the coefficients of A are in F_p, we generalize this criterion to describe when  h(B(x))A(x)+g(B(x))  permutes F_q. This last result generalizes two more constructions from Marcos's paper.

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