A note on complete polynomials over finite fields and their applications in cryptography

A recursive construction of complete mappings over finite fields is provided in this work. These permutation polynomials, characterized by the property that both f(x)∈Fq[x]f(x)∈Fq[x] and its associated mapping f(x)+xf(x)+x are permutations, have an important application in cryptography in the construction of bent–negabent functions which actually leads to some new classes of these functions. Furthermore, we also provide a recursive construction of mappings over finite fields of odd characteristic, having an interesting property that both f(x)f(x) and f(x+c)+f(x)f(x+c)+f(x) are permutations for every c∈Fqc∈Fq. Both the multivariate and univariate representations are treated and some results concerning fixed points and the cycle structure of these permutations are given. Finally, we utilize our main result for the construction of so-called negabent functions and bent functions over finite fields.