Single-Shuffle Full-Open Card-Based Protocols for Any Function

TL;DR

This paper proves that any function can be securely computed using a simple card-based protocol with a single shuffle and open cards, expanding the class of functions feasible for such protocols.

Contribution

It introduces the first proof that all functions admit single-shuffle full-open protocols and offers two constructions balancing card count and shuffle complexity.

Findings

All functions have a single-shuffle full-open protocol.

New constructions enable trade-offs between cards and shuffle complexity.

Variants reduce shuffle complexity by revealing only some cards.

Abstract

A card-based secure computation protocol is a method for $n$ parties to compute a function $f$ on their private inputs $(x_1,\ldots,x_n)$ using physical playing cards, in such a way that the suits of revealed cards leak no information beyond the value of $f(x_1,\ldots,x_n)$. A \textit{single-shuffle full-open} protocol is a minimal model of card-based secure computation in which, after the parties place face-down cards representing their inputs, a single shuffle operation is performed and then all cards are opened to derive the output. Despite the simplicity of this model, the class of functions known to admit single-shuffle full-open protocols has been limited to a few small examples. In this work, we prove for the first time that every function admits a single-shuffle full-open protocol. We present two constructions that offer a trade-off between the number of cards and the complexity of the shuffle operation. These feasibility results are derived from a novel connection between single-shuffle full-open protocols and a cryptographic primitive known as \textit{Private Simultaneous Messages} protocols, which has rarely been studied in the context of card-based cryptography. We also present variants of single-shuffle protocols in which only a subset of cards are revealed. These protocols reduce the complexity of the shuffle operation compared to existing protocols in the same setting.