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

Dillon observed that an APN function F over \({{\mathbb {F}}_{2}^{n}}\) with n greater than 2 must satisfy the condition \(\{F(x) + F(y) + F(z) + F(x + y + z) :\, x,y,z \in {\mathbb {F}}_{2}^{n}\}= {\mathbb {F}}_{2}^{n}\). Recently, Taniguchi (Cryptogr. Commun. 15, 627–647 2023) generalized this condition to functions defined from \({{\mathbb {F}}_{2}^{n}}\) to \({{\mathbb {F}}_{2}^{m}}\), 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 \({{\mathbb {F}}_{2}^{n}}\) to \({{\mathbb {F}}_{2}^{n+1}}\) satisfying this property. In this work, we further study the D-property for (n, m)-functions with \(m\ge n\). 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.