Kasami Power Functions, Permutation Polynomials and Cyclic Difference Sets

Abstract

We study permutation polynomials on F _2nwhich are associated with Kasami power functionsx ^di.e. d = 2^2k — 2^k + 1 fork < nwith gcd(k, n) = 1. We describe in detail the equivalence of a class of permutation polynomials (say “Kasami” permutation polynomials), considered to derive the APN property of Kasami power functions, and the well-known class of MCM permutation polynomials. Explicit and recursive formulae for the polynomial representations of the inverses of Kasami and MCM permutation polynomials are given. As an application the imageBunder the two-to-one mapping (x + 1)^d + x ^d + 1 can be characterized by a trace condition, and the 2-rank ofB* = B\{0}can be determined. We conjecture thatB*is a cyclic difference set, or in other terms that the characteristic sequence ofL \ Bhas ideal autocorrelation. (This conjecture has recently been confirmed, see “Notes added in proof”.)