Componentwise APNness, Walsh uniformity of APN functions, and cyclic-additive difference sets

Abstract

In [Characterizations of the differential uniformity of vectorial functions by the Walsh transform, IEEE Transactions on Information Theory 2017], the author has characterized differentially δ-uniform functions by equalities satisfied by their Walsh transforms. This generalizes the characterization of APN functions by the fourth moment of the Walsh transform. We study two notions which are related: (1) the componentwise APNness (CAPNness) of (n,n)-functions, which is a stronger version of APNness, related to the characterization by the fourth moment, in which the arithmetic mean of WF4(u,v) when u ranges over F2n and v is fixed nonzero in F2n equals 22n+1 (2) the componentwise Walsh uniformity (CWU) of (n,m)-functions (m=n, resp. m=n−1), which is a stronger version of APNness (resp. of differential 4-uniformity) related to one of the new characterizations, in which the arithmetic mean of WF2(u1,v1)WF2(u2,v2)WF2(u1+u2,v1+v2) when u1,u2 range independently over F2n and v1,v2 are fixed nonzero and distinct in F2m, equals 23n. Concerning the first notion, it is known from Berger, Canteaut, Charpin and Laigle-Chapuy that any plateaued function is CAPN if and only if it is AB and that APN power permutations are CAPN. We prove that CAPN functions can exist only if n is odd; this solves an eleven year old open problem by these authors. Concerning the second notion, we show that any crooked function (and in particular any quadratic APN function) is CWU, but we observe also that other APN functions like Kasami functions and the inverse of one of the Gold APN permutations are CWU for n≤11. We show that the CWUness of APN power permutations is equivalent to a property of ΔF={F(x)+F(x+1)+1;x∈F2n}. This new property, that we call cyclic-additive difference set property, is more complex than the cyclic difference set property (proved in the case of Kasami APN functions by Dillon and Dobbertin). We prove it in the case of the inverse of Gold function. In the case of Kasami functions, we observe that the cyclic-additive property is also true for n≤10 even and we leave the proof of the CWUness and the cyclic-additive property as open problems.