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Changing Points in APN Functions

We investigate the differential properties of a construction in which a given function F : F2n → F2n is modified at K ∈ N points in order to obtain a new function G. This is motivated by the question of determining the minimum Hamming distance between two APN functions and can be seen as a generalization of a previously studied construction in which a given function is modified at a single point. We derive necessary and sufficient conditions which the derivatives of F must satisfy for G to be APN, and use these conditions as the basis for an efficient filtering procedure for searching for APN functions whose value differs from that of a given APN function F at a given set of points. We define a quantity mF related to F counting the number of derivatives of a given type, and derive a lower bound on the distance between an APN function F and its closest APN neighbor in terms of mF . Furthermore, the value mF is shown to be invariant under CCZ-equivalence and easier to compute in the case of quadratic functions. We give a formula for mF in the case of F (x) = x which allows us to express a lower bound on the distance between F (x) and the closest APN function in terms of the dimension n of the underlying field. We observe that this distance tends to infinity with n. We also compute mF and the distance to the closest APN function for a representative F from each of the switching classes over F2n for 4 ≤ n ≤ 8. For a given function F and value v, we describe an efficient method for finding all sets of points {u1, u2, . . . , uK} such that setting G(ui) = F (ui) + v and G(x) = F (x) for x 6= ui is APN.