Cyclic Equalizability of Words and Its Application to Card-Based Cryptography

2025-07-07

AI Summary

This paper investigates the cyclic equivalence problem for binary words of equal length and identical Hamming weight in card-based cryptography—specifically, whether two such words can be rendered cyclically equivalent via synchronized letter insertions. It introduces, for the first time in this domain, the notion of *cyclic equivalence* and proposes the novel property of *cyclic equilibratability*. A theoretical bridge is established between combinatorial word structures and card protocols. Leveraging finite group theory, algebraic combinatorics, and formal word theory, the paper proves that all binary words whose length equals their Hamming weight are cyclically equilibratable. This result furnishes a new algebraic-combinatorial paradigm for information erasure and single-cut fully public protocols, substantially enriching the design toolkit and expanding the dimensions of security analysis in card-based cryptography.

Technology Category

• Game Theory and Economic Paradigms: Equilibrium
• Machine Learning: Probabilistic Circuits and Graphical Models
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

Abstract

Card-based cryptography is a research area to implement cryptographic procedures using a deck of physical cards. In recent years, it has been found to be related to finite group theory and algebraic combinatorics, and is becoming more and more closely connected to the field of mathematics. In this paper, we discuss the relationship between card-based cryptography and combinatorics on words for the first time. In particular, we focus on cyclic equality of words. We say that a set of words are cyclically equalizable if they can be transformed to be cyclically equal by repeated simultaneous insertion of letters. The main result of this paper is to show that two binary words of equal length and equal Hamming weight are cyclically equalizable. As applications of cyclic equalizability to card-based cryptography, we describe its applications to the information erasure problem and to single-cut full-open protocols.

Problem

• Exploring cyclic equality of words in combinatorics
• Linking card-based cryptography to word combinatorics
• Applying cyclic equalizability to cryptographic protocols

Innovation

• Links card-based cryptography with combinatorics on words
• Introduces cyclic equalizability concept for binary words
• Applies cyclic equalizability to information erasure problems