Paper 2024/577

Determination of cryptographic tables and properties related to the revised boomerang and its application to a fundamental S-box

Abstract

In symmetric cryptography, vectorial Boolean functions over finite fields F2n derive strong S-boxes. To this end, the S-box should satisfy a list of tests to resist existing attacks, such as the differential, linear, boomerang, and variants. Several tables are employed to measure an S- box’s resistance, such as the difference distribution table (DDT) and the boomerang connectivity table (BCT). Following the boomerang attacks recently revisited in terms of the boomerang switch effect, with a lustra- tion highlighting the power of this technique, a tool called the Boomerang Difference Table (BDT), an alternative to the classical Boomerang BCT, was introduced. Next, two novel tables have been introduced, namely, the Upper Boomerang Connectivity Table (UBCT) and the Lower Boomerang Connectivity Table (LBCT), which are considered improvements over BCT while allowing systematic evaluation of boomerangs to return over mul- tiple rounds. This paper focuses on the new tools for measuring the revisited version of boomerang attacks and the related tables UBCT, LBCT, as well as the so-called Extended Boomerang Connectivity Table (EBCT). Specifically, we shall study the properties of these novel tools and investigate the corresponding tables. We also study their interconnections, their links to the DDT, and their values for affine equivalent vectorial functions and compositional inverses of permutations of F2n . Moreover, we introduce the concept of the nontrivial boomerang connectivity uniformity and determine the explicit values of all the entries of the EBCT, LBCT, and EBCT for the important cryptographic case of the inverse function.