Card-Based Protocols Imply PSM Protocols

Card-based cryptography is the art of cryptography using a deck of physical cards. While this area is known as a research area of recreational cryptography and is recently paid attention in educational purposes, there is no systematic study of the relationship between card-based cryptography and the other “conventional” cryptography. This paper establishes the first generic conversion from card-based protocols to private simultaneous messages (PSM) protocols, a special kind of secure multiparty computation. Our compiler supports “simple” card-based protocols, which is a natural subclass of finite-runtime protocols. The communication complexity of the resulting PSM protocol depends on how many cards are opened in total in all possible branches of the original card-based protocol. This result shows theoretical importance of such “opening complexity” of card-based protocols, which had not been focused in this area. As a consequence, lower bounds for PSM protocols imply those for simple card-based protocols. In particular, if there exists no PSM protocol with subexponential communication complexity for a function f, then there exists no simple card-based protocol with subexponential opening complexity for the same f.