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 ∈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.