A construction of differentially 4-uniform functions from commutative semifields of characteristic 2

Abstract

We construct differentially 4-uniform functions over GF(2n) through Albert's finite commutative semifields, if n is even. The key observation there is that for some k with 0 ≤ k ≤ n - 1, the function fk(x): = (x2k+1 + x)/(x2 + x) is a two to one map on a certain subset Dk(n) of GF(2n). We conjecture that fk is two to one on Dk(n) if and only if (n, k) belongs to a certain list. For (n, k) in this list, fk is proved to be two to one. We also prove that if f2 is two to one on D 2(n) then (n, 2) belongs to the list. © Springer-Verlag Berlin Heidelberg 2007.