The revised boomerang connectivity tables and their connection to the Difference Distribution Table

TL;DR

This paper explores new boomerang connectivity tables (EBCT, LBCT, UBCT) and links them to the Difference Distribution Table, analyzing their effectiveness in assessing the security of certain cryptographic functions.

Contribution

It connects new boomerang tables to the DDT and computes their entries for specific power permutations, enhancing cryptographic security analysis methods.

Findings

Derived explicit formulas for EBCT, LBCT, UBCT entries of Gold, Kasami, Bracken-Leander functions.

Connected boomerang tables to the DDT for differentially δ-uniform functions.

Reproduced known results efficiently as byproducts of the approach.

Abstract

It is well-known that functions over finite fields play a crucial role in designing substitution boxes (S-boxes) in modern block ciphers. In order to analyze the security of an S-box, recently, three new tables have been introduced: the Extended Boomerang Connectivity Table (EBCT), the Lower Boomerang Connectivity Table (LBCT), and the Upper Boomerang Connectivity Table (UBCT). In fact, these tables offer improved methods over the usual Boomerang Connectivity Table (BCT) for analyzing the security of S-boxes against boomerang-style attacks. Here, we put in context these new EBCT, LBCT, and UBCT concepts by connecting them to the DDT for a differentially $\delta$-uniform function and also determine the EBCT, LBCT, and UBCT entries of three classes of differentially $4$-uniform power permutations, namely, Gold, Kasami and Bracken-Leander. We also determine the Double Boomerang Connectivity Table (DBCT) entries of the Gold function. As byproducts of our approach, we obtain some previously published results quite easily.