Paper 2021/1387

Triplicate functions

Lilya Budaghyan, Ivana Ivkovic, and Nikolay Kaleyski

Abstract

We define the class of triplicate functions as a generalization of 3-to-1 functions over GF(2^n) for even values of n. We investigate the properties and behavior of triplicate functions, and of 3-to-1 among triplicate functions, with particular attention to the conditions under which such functions can be APN. We compute the exact number of distinct differential sets of power APN functions and quadratic 3-to-1 functions; we show that, in this sense, quadratic 3-to-1 functions are a generalization of quadratic power APN functions for even dimensions, while quadratic APN permutations are generalizations of quadratic power APN functions for odd dimensions. We show that quadratic 3-to-1 APN functions cannot be CCZ-equivalent to permutations in the case of doubly-even dimensions. We survey all known infinite families of APN functions with respect to the presence of 3-to-1 functions among them, and conclude that for even n almost all of the known infinite families contain functions that are quadratic 3-to-1 or EA-equivalent to quadratic 3-to-1 functions. We also give a simpler univariate representation of the family recently introduced by Gologlu singly-even dimensions n than the ones currently available in the literature.