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.

翻译：循环可等化性是Shinagawa与Nuida于2025年在纸牌密码学研究中提出的概念。非正式地说，一组单词被称为循环可等化，若通过在所有单词的相同位置插入相同字母，可将其转化为彼此互为循环移位的单词。Shinagawa与Nuida证明：两个等长二进制单词循环可等化当且仅当它们具有相同的汉明重量。他们还提出了更大字母表上循环可等化性的刻画问题。本文通过证明两个单词循环可等化当且仅当它们具有相同的帕里克向量，完整刻画了任意有限字母表上两个单词的循环可等化性。