  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.
