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

TL;DR

This paper explores the concept of cyclic equalizability of words, establishing a key property for binary words, and applies this to enhance protocols in card-based cryptography, especially for information erasure and secure card operations.

Contribution

It introduces the concept of cyclic equalizability of words and proves a new property for binary words, connecting combinatorics on words with cryptographic applications.

Findings

Binary words of equal length and Hamming weight are cyclically equalizable.

Cyclic equalizability aids in solving information erasure problems.

Applications to single-cut full-open protocols in card cryptography.

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.