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

2024-07-17

arXiv.org

AI Summary

This work addresses the security evaluation of S-boxes against boomerang-style attacks, focusing on establishing rigorous theoretical connections among boomerang connectivity tables (BCTs) and the differential distribution table (DDT). Method: Leveraging finite field theory and combinatorial counting techniques, we systematically develop extended BCT variants—namely the extended BCT (EBCT), lower BCT (LBCT), and upper BCT (UBCT)—and derive their closed-form expressions for three classes of 4-uniform power permutations: Gold, Kasami, and Bracken–Leander functions. We further introduce and compute the double boomerang connectivity table (DBCT), obtaining its first closed-form solution for Gold functions. Contribution/Results: Our analysis uncovers deep algebraic links between the BCT family and δ-uniformity, unifies the mathematical relationships among multiple BCT variants and the DDT, simplifies and reproves several existing results, and provides novel theoretical tools for assessing S-box resistance to boomerang attacks.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Combinatorial Optimization
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

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.

Problem

• S-box Security
• Cryptographic Attacks
• Differential Distribution Table

Innovation

• Differential Distribution Table
• Updated Boomerang Connectivity Table
• 4-Uniform Power Permutations