On a construction of quadratic APN functions
Lilya Budaghyan, Claude Carlet, Gregor Leander · 2009
In a recent paper, the authors introduced a method for constructing new quadratic APN functions from known ones. Applying this method, they obtained the function x3+ trn(x9) which is APN over F2nfor any positive integer n. The present paper is a continuation of this work. We give sufficient conditions on linear functions L1and L2from F2nto itself such that the function L1(x3) + L2(x9) is APN over F2n. We show that this can lead to many new cases of APN functions. In particular, we get two families of APN functions x3+ a-1tr3n(a3x9+ a6x18) and x3+ a-1tr3n(a6x18+ a12x36) over F2nfor any n divisible by 3 and a ϵ F2n*. We prove that for n = 9, these families are pairwise different and differ from all previously known families of APN functions, up to the most general equivalence notion, the CCZ-equivalence. We also investigate further sufficient conditions under which the conditions on the linear functions L1and L2are satisfied.