AI 中文总结
研究在二进制字中仅插入0使序列成为循环移位的0循环可均等性问题，通过将二进制字编码到四字母字母表，把可均等性简化为更简单条件并显式构建插入，证明等长二进制字0循环可均等当且仅当汉明重量相等。
AI 中文摘要
随机切牌是基于卡片的密码学中最基本的洗牌操作之一：它将一叠面朝下的牌按秘密数量旋转。在这种洗牌操作下，当且仅当两个牌序列是彼此的循环移位时，它们才无法区分。这引发了一个问题：给定两个牌序列，在匹配位置插入牌能否使它们无法区分？先前研究表明，只要两个字是彼此的排列，当可以插入任何牌时，这种插入总是可行的。本文考虑更强的限制：如果牌是二进制的，只携带0或1，我们能否只插入0使序列无法区分？如果可以在两个序列的匹配位置插入0，使得到的字是彼此的循环移位，我们称这两个字是0循环可均等的。我们的主要结果是，等长的两个二进制字是0循环可均等的，当且仅当它们具有相等的汉明重量，即相同数量的1位。由于相等的汉明重量显然是必要的，本文的内容是证明它也是充分的。我们的证明是构造性的：我们将一对二进制字编码为一个在四字母字母表{A, B, X, O}上的单个字，将可均等性简化为这种编码中的一个更简单的条件，并明确构建所需的插入。
英文摘要
The random cut is one of the most fundamental shuffles in card-based cryptography: it rotates a sequence of face-down cards by a secret amount. Under this shuffle, two sequences of cards are indistinguishable if and only if they are cyclic shifts of each other. This motivates the question of whether, given two sequences of cards, inserting cards at matching positions can make them indistinguishable. A previous study shows that such an insertion is always possible when any cards may be inserted, as long as the two words are permutations of each other. This paper considers a stronger restriction: if the cards are binary, carrying only 0 or 1, can we insert only 0s to make the sequences indistinguishable? We call two words 0-cyclically equalizable if one can insert 0s into both sequences at matching positions so that the resulting words are cyclic shifts of each other. Our main result is that two binary words of equal length are 0-cyclically equalizable if and only if they have equal Hamming weight, that is, the same number of 1-bits. Since equal Hamming weight is clearly necessary, the content of the paper is to show that it is also sufficient. Our proof is constructive: we encode a pair of binary words as a single word over the four-letter alphabet {A, B, X, O}, reduce equalizability to a simpler condition in this encoding, and build the required insertion explicitly.
Comments20 pages, 5 figures. Code available at https://github.com/SThongjarast/Binary-Equalization
