Some results concerning cryptographically significant mappings over GF(2n)

In this paper we investigate the existence of permutation polynomials of the form F(x) = xd + L(x) over GF(2n), L being a linear polynomial. The results we derive have a certain impact on the long-term open problem on the nonexistence of APN permutations over GF(2n), when n is even. It is shown that certain choices of exponent d cannot yield APN permutations for even n. When n is odd, an infinite class of APN permutations may be derived from Gold mapping x3 in a recursive manner, that is starting with a specific APN permutation on GF(2k), k odd, APN permutations are derived over GF(2k+2i) for any i ≥ 1. But it is demonstrated that these classes of functions are simply affine permutations of the inverse coset of the Gold mapping x3. This essentially excludes the possibility of deriving new EA-inequivalent classes of APN functions by applying the method of Berveglieri et al. (approach proposed at Asiacrypt 2004, see [3]) to arbitrary APN functions.