The revised boomerang connectivity tables and their connection to the Difference Distribution Table
Kirpa Garg, Sartaj Ul Hasan, Constanza Riera, Pantelimon Stanica
TL;DR
This work establishes concrete links between Extended, Lower, and Upper Boomerang Connectivity Tables ($EBCT$, $LBCT$, $UBCT$) and the Difference Distribution Table ($DDT$) for $\delta$-uniform functions, enabling explicit boomerang-connectivity analysis from $DDT$ data. It develops general, inverse-free formulas for $EBCT$ and extends the definitions to all functions, then studies invariance under CCZ, EA, and affine-equivalence, highlighting both preserved and non-preserved aspects. The authors derive exact $EBCT$, $LBCT$, and $UBCT$ entries for three prominent 4-differentially uniform power permutations (Gold, Kasami, Bracken-Leander) and compute the $DBCT$ for the Gold function, offering new, replicable tables and byproducts that recover known results in special cases. Overall, the paper provides a practical framework to assess boomerang-resistance in S-box design by translating boomerang-connectivity statistics into $DDT$-driven computations, with clear avenues for further evaluation of additional 4-differential uniform functions.
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 $δ$-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.