Impossibility Results of Card-Based Protocols via Mathematical Optimization

2025-09-22

AI Summary

This work addresses the expressiveness boundary of the Single-Cut-For-Opening (SCFO) protocol in card-based cryptography. We establish, for the first time, a general impossibility framework applicable to an arbitrary number of cards—requiring only that auxiliary cards share the same color. By modeling SCFO protocols as joint optimization problems over Boolean functions and combinatorial constraints, and leveraging mathematical programming together with formal derivation, we rigorously characterize their feasibility conditions. Our main result proves that, under the constraint of monochromatic auxiliary cards, no novel three-variable Boolean function realizable via SCFO can surpass the capabilities of existing constructions. This breakthrough transcends prior analyses limited to small card counts and establishes a fundamental upper bound on SCFO expressiveness. To our knowledge, this is the first universal negative result for card-based cryptographic protocols, providing a foundational limit on what can be securely computed using SCFO.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Search and Optimization: Combinatorial Optimization

Application Category

• Security and Privacy: Applications of cryptography
• Responsible Web: Measurement, analysis, and circumvention of Web censorship
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

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.

Problem

• Proving impossibility results for card-based cryptographic protocols
• Establishing limitations of single-cut full-open protocols for Boolean functions
• Developing optimization-based framework valid for any number of cards

Innovation

• Mathematical optimization enables impossibility proofs
• Proofs hold for large number of cards
• No new SCFO protocols for Boolean functions