In card-based cryptography, designing AND protocols in committed format is a major research topic. The state-of-the-art AND protocol proposed by Koch, Walzer, and Härtel in ASIACRYPT 2015 uses only four cards, which is the minimum permissible number. The minimality of their protocol relies on somewhat complicated shuffles having non-uniform probabilities of possible outcomes. Restricting the allowed shuffles to uniform closed ones entails that, to the best of our knowledge, six cards are sufficient: the six-card AND protocol proposed by Mizuki and Sone in 2009 utilizes the random bisection cut, which is a uniform and cyclic (and hence, closed) shuffle. Thus, a question has arisen: “Can we improve upon this six-card protocol using only uniform closed shuffles?” In other words, the existence or otherwise of a five-card AND protocol in committed format using only uniform closed shuffles has been one of the most important open questions in this field. In this paper, we answer the question affirmatively by designing five-card committed-format AND protocols using only uniform cyclic shuffles. The shuffles that our protocols use are the random cut and random bisection cut, both of which are uniform cyclic shuffles and can be easily implemented by humans.

在基于卡的加密中，以提交格式设计AND协议是一个主要的研究主题。由Koch，Walzer和Härtel在ASIACRYPT 2015中提出的最新AND协议仅使用四张卡，这是允许的最小数量。他们协议的最小性依赖于一些复杂的改组，这些改组具有可能结果的不均匀概率。据我们所知，将允许的改组限制为统一的封闭改组意味着六张卡就足够了：Mizuki和Sone在2009年提出的六张卡AND协议利用了随机等分分割，这是统一且循环的（因此，关闭）随机播放。因此，出现了一个问题：“我们是否可以仅使用统一的封闭洗牌来改进此六张卡协议？” 换一种说法，仅使用统一的封闭混洗，是否存在承诺格式的五卡AND协议一直是该领域最重要的开放性问题之一。在本文中，我们通过仅使用统一循环混洗设计五张卡提交格式AND协议来肯定地回答这个问题。我们的协议使用的改组是随机剪切和随机二等分剪切，它们都是统一的循环改组，并且可以由人类轻松实现。