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Quantum Secret Sharing Scheme on Hypercyclic Quantum Structures

Lei Li, Zhi Li

TL;DR

The paper addresses the challenge of designing efficient quantum secret sharing for irregular hypergraph access structures, focusing on hypercycles with three hyperedges. It proves a complete classification of these hypercycles into 12 isomorphism types and shows that some are ideal (rate 1) while others achieve a rate of 2/3, then constructs classical secret sharing schemes with optimal information rates by combining Simmons' geometric method with Shamir's threshold scheme and subsequently realizes perfect QSS using single photons. A core contribution is the introduction of the idealized information rate and its relation to the efficiency of minimally authorized subsets, with proofs that the proposed QSS schemes attain the highest achievable efficiency among known solutions. The work also provides explicit constructions and rates for 83 hypercycle-based access structures with seven participants, offering practical guidance for implementing high-efficiency QSS in quantum networks.

Abstract

This paper investigates the construction of efficient quantum secret sharing schemes for quantum access structures based on hypergraphs with three hyperedges. We prove that hypercycles with three hyperedges are quantum access structures if and only if they can be classified into 12 non-isomorphic types under hypergraph isomorphism, and these hypercyclic quantum access structures encompass all four hyperstars with three hyperedges.In prior work, efficient and perfect quantum secret sharing schemes were constructed for hyperstar access structures with three hyperedges using single photons in d -dimensional quantum systems. These schemes correspond to classical perfect secret sharing schemes with optimal information rates. However, for hypercyclic access structures with three hyperedges, the method described above fails to yield classical perfect secret sharing schemes with optimal information rates. In this work, we explicitly construct classical perfect secret sharing schemes with optimal information rates for these access structures by combining Simmons' geometric method with Shamir's threshold scheme. Subsequently, we employ single photons in d-dimensional quantum systems to build perfect QSS schemes upon these classical constructions.We introduce the concept of idealized information rate for minimal access structures in perfect QSS schemes. By extending the notion of efficiency proposed by Cabello [Phys. Rev. Lett., vol. 85, no. 26, p. 5635, 2000] for quantum key distribution protocols to QSS, we define the efficiency of minimally authorized subsets. Furthermore, we establish the relationship between the efficiency of minimally authorized subsets and the idealized information rate of minimal access structures. Finally, we rigorously prove that the QSS schemes constructed in this work achieve the highest efficiency among existing solutions.

Quantum Secret Sharing Scheme on Hypercyclic Quantum Structures

TL;DR

The paper addresses the challenge of designing efficient quantum secret sharing for irregular hypergraph access structures, focusing on hypercycles with three hyperedges. It proves a complete classification of these hypercycles into 12 isomorphism types and shows that some are ideal (rate 1) while others achieve a rate of 2/3, then constructs classical secret sharing schemes with optimal information rates by combining Simmons' geometric method with Shamir's threshold scheme and subsequently realizes perfect QSS using single photons. A core contribution is the introduction of the idealized information rate and its relation to the efficiency of minimally authorized subsets, with proofs that the proposed QSS schemes attain the highest achievable efficiency among known solutions. The work also provides explicit constructions and rates for 83 hypercycle-based access structures with seven participants, offering practical guidance for implementing high-efficiency QSS in quantum networks.

Abstract

This paper investigates the construction of efficient quantum secret sharing schemes for quantum access structures based on hypergraphs with three hyperedges. We prove that hypercycles with three hyperedges are quantum access structures if and only if they can be classified into 12 non-isomorphic types under hypergraph isomorphism, and these hypercyclic quantum access structures encompass all four hyperstars with three hyperedges.In prior work, efficient and perfect quantum secret sharing schemes were constructed for hyperstar access structures with three hyperedges using single photons in d -dimensional quantum systems. These schemes correspond to classical perfect secret sharing schemes with optimal information rates. However, for hypercyclic access structures with three hyperedges, the method described above fails to yield classical perfect secret sharing schemes with optimal information rates. In this work, we explicitly construct classical perfect secret sharing schemes with optimal information rates for these access structures by combining Simmons' geometric method with Shamir's threshold scheme. Subsequently, we employ single photons in d-dimensional quantum systems to build perfect QSS schemes upon these classical constructions.We introduce the concept of idealized information rate for minimal access structures in perfect QSS schemes. By extending the notion of efficiency proposed by Cabello [Phys. Rev. Lett., vol. 85, no. 26, p. 5635, 2000] for quantum key distribution protocols to QSS, we define the efficiency of minimally authorized subsets. Furthermore, we establish the relationship between the efficiency of minimally authorized subsets and the idealized information rate of minimal access structures. Finally, we rigorously prove that the QSS schemes constructed in this work achieve the highest efficiency among existing solutions.

Paper Structure

This paper contains 20 sections, 8 theorems, 70 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Discussion
  4. Contributions
  5. Organization
  6. BACKGROUND
  7. Notation
  8. Hypergraph Access Structure
  9. Perfect Secret Sharing Scheme
  10. Mutually Unbiased Basis
  11. Simmons 's Geometry Construction Method
  12. hypercycle access structure with Three Hyperedges and Its Optimal Information Rate
  13. Types of hypercycle access structures with three hyper-edges
  14. The optimal information rate of hypercycle structures with 3 hyperedges
  15. The QSS scheme based on the hypercycle access structure containing three hyperedges
  16. ...and 5 more sections

Key Result

Lemma 2.1

Assume that $\varGamma$ is an access structure with minimal access structure $\varGamma_0$ref48. $l\geq 1$ is an integer. Let $K$ be a specified key set and let $D_h=\{\varGamma_{h,1},\varGamma_{h,2},\cdot,\varGamma_{h,n_h}\}$ be an ideal decomposition of $\varGamma_0$ for the key set $K$, where $1\ Then there exists a perfect secret sharing scheme that implements $\varGamma$ with an information r

Figures (4)

  • Figure 1: The hypergraph corresponding to minimal access structure $\varGamma^{(2)}_0$.
  • Figure 2: Schematic diagram of the parameters of the secret share on the minimal access structure $\varGamma^{(3)}_0$, where $\varPi$ stands for the blue plane.
  • Figure 3: Hypercycle quantum access structure containing 3 hyperedges, $H_3(P,\varGamma_i)(i\in[4])$ is a hyperstar.
  • Figure 4: Schematic diagram of the participant in the minimal authorized subset $A_!A_2A_4$ recovering the key $(s,k)$

Theorems & Definitions (26)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Lemma 2.1
  • Lemma 2.2
  • Theorem 3.1
  • ...and 16 more