Impossibility Results of Card-Based Protocols via Mathematical Optimization

Abstract: This paper introduces mathematical optimization as a new method for proving impossibility proofs in the field of card-based cryptography. While previous impossibility proofs were often limited to cases involving a small number of cards, this new approach establishes results that hold for a large number of cards. The research focuses on single-cut full-open (SCFO) protocols, which consist of performing one random cut and then revealing all cards. The main contribution is that for any three-variable Boolean function, no new SCFO protocols exist beyond those already known, under the condition that all additional cards have the same color. The significance of this work is that it provides a new framework for impossibility proofs and delivers a proof that is valid for any number of cards, as long as all additional cards have the same color.