Impossibility Results of Card-Based Protocols via Mathematical Optimization

TL;DR

The paper tackles the problem of proving impossibility results in card-based cryptography by introducing a mathematical optimization approach for single-cut full-open (SCFO) protocols. It formulates the constructibility of $(*,0)$-SCFO (and, by symmetry, $(0,*)$-SCFO) as an integer optimization problem and develops a search algorithm to test all input permutations and insertions. Applying this method to all $3$-variable Boolean functions, the authors show no new $(*,0)$-SCFO protocols exist except for the equality function, illustrating the framework’s effectiveness in producing large-card impossibility results. The work demonstrates the potential of optimization techniques to advance card-based cryptography, while also outlining significant open questions, such as extending to four variables and general SCFO protocol classes.

Abstract

This paper introduces mathematical optimization as a new method for proving impossibility results 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 proving impossibility results and delivers a proof that is valid for any number of cards, as long as all additional cards have the same color.