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

Masahito Hayashi, Seunghoan Song

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.

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

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.
Paper Structure (38 sections, 42 theorems, 51 equations, 4 figures, 7 tables)

This paper contains 38 sections, 42 theorems, 51 equations, 4 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Vector space and matrix over finite field
  4. Fundamentals of Quantum Information Theory
  5. Stabilizer formalism over finite fields
  6. Information quantities
  7. Access structure
  8. Our models
  9. Formulation of quantum versions of SS protocols
  10. Formulation of quantum versions of SPIR protocol
  11. Classical linear protocols
  12. Linear CSS
  13. Multi-target monotone span program (MMSP)
  14. Linear CSPIR
  15. Linear quantum protocols without preshared entanglement with user
  16. ...and 23 more sections

Key Result

Lemma 1

The following conditions are equivalent for an ${\bar{\mathsf{n}}}\times (\mathsf{y}+\mathsf{x})$ matrix $(G,F)$ and a subset $\mathcal{A}\subset [{\bar{\mathsf{n}}}]$, Also, the following conditions are equivalent for a ${\bar{\mathsf{n}}}\times (\mathsf{y}+\mathsf{x})$ matrix $(G,F)$ and a subset $\mathcal{B}\subset [{\bar{\mathsf{n}}}]$.

Figures (4)

  • Figure 1: Quantum SS protocols where the end-user receives the shares from Player 2 and Player 3. Fig. (a), (b), and (c) show a CQSS protocol, a QQSS protocol, and an EASS protocol, respectively. The notations in the above figures will be defined in Section \ref{['S4']}.
  • Figure 2: Classical-quantum (CQ) SPIR protocols where Sever 1 and Server 2 collude and Server 2 and Server 3 respond to the user.
  • Figure 3: Entanglement-assisted (EA) SPIR protocols where Sever 1 and Server 2 collude and Server 2 and Server 3 respond to the user.
  • Figure 4: One-to-one relations among various linear protocols. In this figure, the word "linear" is omitted. Arrows of each color show a one-to-one relation among several protocols. $\hbox{\textcircled{\scriptsize{1}}}$ shows the restriction that $G=(G^{(1)},G^{(2)})$ and $G^{(1)}$ is self-column-orthogonal. $\hbox{\textcircled{\scriptsize{2}}}$ shows the restriction that $G^{(2)}=\emptyset$ and $F$ is column-orthogonal to $G^{(1)}$.

Theorems & Definitions (86)

  • Example 1
  • Definition 1: $(\mathfrak{A},\mathfrak{B})$-security
  • Definition 2: $(\mathfrak{A},\mathfrak{B})$-QQSS
  • Definition 3
  • Definition 4: Linear CSS
  • Definition 5: $({\bar{\mathsf{n}}},\mathsf{x})$-MDS code
  • Remark 1
  • Lemma 1
  • proof
  • Definition 6: Multi-target monotone span program (MMSP)
  • ...and 76 more