Kasami power functions, permutation polynomials and cyclic difference sets

We study permutation polynomials on F-2n, which are associated with Kasami power functions x(d), i.e. d = 2(2k) - 2(k) + 1 for k < n with 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 image beta under the two-to-one mapping (x + 1)(d) + x(d) + 1 can be characterized by a trace condition, and the 2-rank of beta* = beta\{0} can be determined. We conjecture that beta* is a cyclic difference set, of in other terms that the characteristic sequence of L \ beta has ideal autocorrelation. (This conjecture has recently been confirmed, see "Notes added in proof".).