2023/02/27 by Abbondati, Matteo, Calderini, Marco, Villa, Irene
#Combinatorics (math.CO) #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT)
paper · doi:10.48550/arxiv.2302.13922
Dillon observed that an APN function F over \mathbbF2n with n greater than 2 must satisfy the condition \F(x) + F(y) + F(z) + F(x + y + z) : x,y,z ∈\mathbbF2n\= \mathbbF2n. Recently, Taniguchi (2023) generalized this condition to functions defined from \mathbbF2n to \mathbbF2m, 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 \mathbbF2n to \mathbbF2n+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.