Some results concerning cryptographically significant mappings over GF(2 (n) )
In this paper we investigate the existence of permutation polynomials of the form F(x) = x (d) + L(x) over GF(2 (n) ), 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(2 (n) ), 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 x (3) in a recursive manner, that is starting with a specific APN permutation on GF(2 (k) ), k odd, APN permutations are derived over GF(2 (k+2i) ) for any i a parts per thousand yen 1. But it is demonstrated that these classes of functions are simply affine permutations of the inverse coset of the Gold mapping x (3). 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.
permutation polynomials apn functions power mappings s-box ea-equivalence ccz-equivalence