Unified Approach to Secret Sharing and Symmetric Private Information Retrieval with Colluding Servers in Quantum Systems

TL;DR

This paper unifies quantum secret sharing and symmetric private information retrieval using multi-target monotone span programs, introducing a new entanglement-assisted setting and characterizing various quantum protocols.

Contribution

It introduces the entanglement-assisted setting for quantum secret sharing and links it to MMSP, providing a unified framework for quantum SS and SPIR protocols.

Findings

Quantum secret sharing with EA setting linked to MMSP

Characterization of linear quantum SS protocols via MMSP

Introduction of quantum maximum distance separable codes

Abstract

This paper unifiedly addresses two kinds of key quantum secure tasks, i.e., quantum versions of secret sharing (SS) and symmetric private information retrieval (SPIR) by using multi-target monotone span program (MMSP), which characterizes the classical linear protocols of SS and SPIR. SS has two quantum extensions; One is the classical-quantum (CQ) setting, in which the secret to be sent is classical information and the shares are quantum systems. The other is the quantum-quantum (QQ) setting, in which the secret to be sent is a quantum state and the shares are quantum systems. The relation between these quantum protocols and MMSP has not been studied sufficiently. We newly introduce the third setting, i.e., the entanglement-assisted (EA) setting, which is defined by modifying the CQ setting with allowing prior entanglement between the dealer and the end-user who recovers the secret by collecting the shares. Showing that the linear version of SS with the EA setting is directly linked to MMSP, we characterize linear quantum versions of SS with the CQ ad QQ settings via MMSP. Further, we introduce the EA setting of SPIR, which is shown to link to MMSP. In addition, we discuss the quantum version of maximum distance separable codes.