Paper 2021/1387
Triplicate functions
Lilya Budaghyan and Ivana Ivkovic and Nikolay Kaleyski
Abstract
We define the class of triplicate functions as a generalization of 3-to-1 functions over the finite field F 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, quadratic 3-to-1 functions, and quadratic APN permutations; 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 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. Using the developed framework, we give the first proof that the infinite APN families of Budaghyan, Helleseth and Kaleyski; of Gologlu; and of Zheng, Kan, Li, Peng, and Tang have a Gold-like Walsh spectrum. We also give a simpler univariate representation of the Gogloglu family for dimensions n = 2m with m odd than the ones currently available in the literature. We conduct a computational search for quadratic 3-to-1 functions in even dimensions n ≤ 12. We find one new APN instance for n = 8, six new APN instances for n = 10, and the first sporadic APN instance for n = 12 since 2006. We provide a list of all known 3-to-1 APN functions for n ≤ 12.
Metadata
- Available format(s)
- Category
- Foundations
- Publication info
- Preprint. MINOR revision.
- Keywords
- boolean functionsAPN functionsdifferential uniformitydifferential sets
- Contact author(s)
-
Ivana Ivkovic @ student uib no
Nikolay Kaleyski @ uib no - History
- 2021-12-26: revised
- 2021-10-15: received
- See all versions
- Short URL
- https://ia.cr/2021/1387
- License
-
CC BY