Cyclic Equalizability Characterized by Parikh Vectors
Sarunyu Thongjarast, Sarit Pasiphol, Suthee Ruangwises
TLDR
This paper shows two words are cyclically equalizable over any alphabet if and only if they have the same Parikh vector, solving an open problem.
Key contributions
- Characterizes cyclic equalizability for two words over any finite alphabet.
- Solves an open problem posed by Shinagawa and Nuida in card-based cryptography.
- Proves cyclic equalizability is equivalent to having the same Parikh vector.
- Generalizes prior results that only applied to binary words (Hamming weight).
Why it matters
This paper provides a fundamental characterization of cyclic equalizability, a concept relevant to card-based cryptography. By extending the characterization from binary to arbitrary alphabets using Parikh vectors, it offers a more general and powerful tool for future research and applications.
Original Abstract
Cyclic equalizability is a notion introduced by Shinagawa and Nuida in 2025, in the study of card-based cryptography. Informally, a collection of words is cyclically equalizable if, by inserting the same letters at the same positions in all words, they can be transformed into words that are cyclic shifts of one another. Shinagawa and Nuida showed that two binary words of equal length are cyclically equalizable if and only if they have the same Hamming weight. They also posed the problem of characterizing cyclic equalizability over larger alphabets. In this paper, we completely characterize cyclic equalizability for two words over an arbitrary finite alphabet by proving that two words are cyclically equalizable if and only if they have the same Parikh vector.
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