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In this paper we define a notion of partial APNness and find various characterizations and constructions of classes of functions satisfying this condition. We connect this notion to the known conjecture that APN functions modified at a point cannot remain APN. In the second part of the paper, we find conditions for some transformations not to be partially APN, and in the process, we find classes of functions that are never APN for infinitely many extensions of the prime field F 2 , extending some earlier results of Leander and Rodier.

In this paper we investigate several families of monomial functions with APN-like exponents that are not APN, but are partially 0-APN for infinitely many extensions of the binary field F 2 . We also investigate the differential uniformity of some binomial partial APN functions. Furthermore, the partial APN-ness for some classes of multinomial functions is investigated. We show also that the size of the pAPN spectrum is preserved under CCZ-equivalence.

We investigate the differential properties of a vectorial Boolean function G obtained by modifying an APN function F . This generalizes previous constructions where a function is modified at a few points. We characterize the APN-ness of G via the derivatives of F , and deduce an algorithm for searching for APN functions whose values differ from those of F only on a given U ⊆ F 2 n .We introduce a value Π F associated with any F , which is invariant under CCZ-equivalence. We express a lower bound on the distance between a given APN function F and the closest APN function in terms of Π F . We show how Π F can be computed efficiently for F quadratic. We compute Π F for all known APN functions over F 2 n up to n ≤ 8. This is the first new CCZ-invariant for APN functions to be introduced within the last ten years.We derive a mathematical formula for this lower bound for the Gold function F (x) = x 3 , and observe that it tends to infinity with n. Finally, we describe how to efficiently find all sets U such that taking G(x) = F (x) + v for x ∈ U and G(x) = F (x) for x / ∈ U is APN.

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