The second-order zero differential uniformity of the swapped inverse functions over finite fields

TL;DR

This paper studies the second-order zero differential uniformity of swapped inverse functions over finite fields, providing new insights into their cryptographic properties and characterizing classes with uniformity equal to 4 in even characteristic.

Contribution

It introduces the first analysis of second-order zero differential uniformity for swapped inverse functions, especially for non-power functions in even characteristic.

Findings

Characterized second-order zero differential spectrum for certain swapped inverse functions.

Identified classes of non-power functions with uniformity equal to 4.

Extended understanding of cryptographic properties of swapped inverse functions.

Abstract

The Feistel Boomerang Connectivity Table (FBCT) was proposed as the feistel counterpart of the Boomerang Connectivity Table. The entries of the FBCT are actually related to the second-order zero differential spectrum. Recently, several results on the second-order zero differential uniformity of some functions were introduced. However, almost all of them were focused on power functions, and there are only few results on non-power functions. In this paper, we investigate the second-order zero differential uniformity of the swapped inverse functions, which are functions obtained from swapping two points in the inverse function. We also present the second-order zero differential spectrum of the swapped inverse functions for certain cases. In particular, this paper is the first result to characterize classes of non-power functions with the second-order zero differential uniformity equal to 4, in even characteristic.