On Dillon’s property of (n, m)-functions

Dillon observed that an APN function F over F-2(n) with n greater than 2 must satisfy the condition {F(x) + F(y) + F(z) + F(x + y + z) :x, y, z is an element of F-2(n)} = F-2(n). Recently, Taniguchi (Cryptogr. Commun. 15, 627-647 2023) generalized this condition to functions defined from F-2(n) to F-2(m), with m > n, calling it the D-property. Taniguchi gave some characterizations of APN functions satisfying the D-property and provided some families of APN functions from F-2(n) to F-2(n+1) satisfying this property. In this work, we further study the D-property for (n, m)-functions with m >= n. We give some combinatorial bounds on the dimension m for the existence of such functions. Then, we characterize the D-property in terms of the Walsh transform and for quadratic functions we give a characterization of this property in terms of the ANF. We also give a simplification on checking the D-property for quadratic functions, which permits to extend some of the APN families provided by Taniguchi. We further focus on the class of the plateaued functions, providing conditions for the D-property. To conclude, we show a connection of some results obtained with the higher-order differentiability and the inverse Fourier transform.