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Asymptotically ideal Disjunctive Hierarchical Secret Sharing Scheme with an Explicit Construction

Jian Ding, Cheng Wang, Haifeng Yu, Hongju Li, Cheng Shu

Abstract

Disjunctive Hierarchical Secret Sharing (DHSS) scheme is a secret sharing scheme in which the set of all participants is partitioned into disjoint subsets. Each disjoint subset is said to be a level, and different levels have different degrees of trust and different thresholds. If the number of cooperating participants from a given level falls to meet its threshold, the shortfall can be compensated by participants from higher levels. Many ideal DHSS schemes have been proposed, but they often suffer from big share sizes. Conversely, existing non-ideal DHSS schemes achieve small share sizes, yet they fail to be both secure and asymptotically ideal simultaneously. In this work, we present an explicit construct of an asymptotically ideal DHSS scheme by using a polynomial, multiple linear homogeneous recurrence relations and one-way functions. Although our scheme has computational security and many public values, it has a small share size and the dealer is required polynomial time.

Asymptotically ideal Disjunctive Hierarchical Secret Sharing Scheme with an Explicit Construction

Abstract

Disjunctive Hierarchical Secret Sharing (DHSS) scheme is a secret sharing scheme in which the set of all participants is partitioned into disjoint subsets. Each disjoint subset is said to be a level, and different levels have different degrees of trust and different thresholds. If the number of cooperating participants from a given level falls to meet its threshold, the shortfall can be compensated by participants from higher levels. Many ideal DHSS schemes have been proposed, but they often suffer from big share sizes. Conversely, existing non-ideal DHSS schemes achieve small share sizes, yet they fail to be both secure and asymptotically ideal simultaneously. In this work, we present an explicit construct of an asymptotically ideal DHSS scheme by using a polynomial, multiple linear homogeneous recurrence relations and one-way functions. Although our scheme has computational security and many public values, it has a small share size and the dealer is required polynomial time.
Paper Structure (10 sections, 6 theorems, 55 equations, 1 table)

This paper contains 10 sections, 6 theorems, 55 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some secret sharing schemes
  4. Linear homogeneous recurrence relations
  5. An asymptotically ideal DHSS scheme
  6. Our scheme
  7. Security analysis of our scheme
  8. Conclusion
  9. Review of GPN scheme
  10. Counterexamples

Key Result

Theorem 1

Let $\alpha_1,\alpha_2,\dots,\alpha_m$ be distinct roots of the auxiliary equation in Definition def: Auxiliary equation with multiplicities $t_1,t_2,\dots,t_m$, respectively. If $\sum_{\ell=1}^{m}t_{\ell}=t$, the general term for the $t$-order LHR relation $(u_i)_{i\geq 0}$ in Definition def: LHR i where polynomials $g_{\ell}(x)=b_{\ell,0}+b_{\ell,1}x+\cdots+b_{\ell, t_{\ell}-1}x^{t_{\ell}-1}\in

Theorems & Definitions (18)

  • Definition 1: Secret sharing scheme
  • Definition 2: Information rate, Ding2023
  • Definition 3: Asymptotically ideal DHSS scheme, Quisquater2002
  • Definition 4: Linear homogeneous recurrence relation, Biggs1989
  • Definition 5: Auxiliary equation,Biggs1989
  • Theorem 1: YuanJiaotong2022
  • Theorem 2: Correctness
  • proof
  • Lemma 1
  • proof
  • ...and 8 more