Generalizations of complete mappings of finite fields and some applications

Permutation polynomials of finite fields have many applications in cryptography, coding theory, and combinatorics. In the first part we mention some applications of a class of permutation polynomials with additional features which includes complete mappings and orthomorphisms. In the second part we discuss several constructions. We focus on linearized polynomials and cyclotomic polynomials of degree 2.

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