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Asymptotically Ideal Hierarchical Secret Sharing Based on CRT for Integer Ring

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

Abstract

In Shamir's secret sharing scheme, all participants possess equal privileges. However, in many practical scenarios, it is often necessary to assign different levels of authority to different participants. To address this requirement, Hierarchical Secret Sharing (HSS) schemes were developed, which partitioned all participants into multiple subsets and assigned a distinct privilege level to each. Existing Chinese Remainder Theorem (CRT)-based HSS schemes benefit from flexible share sizes, but either exhibit security flaws or have an information rate less than $\frac{1}{2}$. In this work, we propose a disjunctive HSS scheme and a conjunctive HSS scheme by using the CRT for integer ring and one-way functions. Both schemes are asymptotically ideal and are proven to be secure.

Asymptotically Ideal Hierarchical Secret Sharing Based on CRT for Integer Ring

Abstract

In Shamir's secret sharing scheme, all participants possess equal privileges. However, in many practical scenarios, it is often necessary to assign different levels of authority to different participants. To address this requirement, Hierarchical Secret Sharing (HSS) schemes were developed, which partitioned all participants into multiple subsets and assigned a distinct privilege level to each. Existing Chinese Remainder Theorem (CRT)-based HSS schemes benefit from flexible share sizes, but either exhibit security flaws or have an information rate less than . In this work, we propose a disjunctive HSS scheme and a conjunctive HSS scheme by using the CRT for integer ring and one-way functions. Both schemes are asymptotically ideal and are proven to be secure.
Paper Structure (11 sections, 10 theorems, 58 equations)

This paper contains 11 sections, 10 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Relevant mathematical tools
  4. Secret sharing
  5. A novel asymptotically ideal DHSS scheme
  6. Our scheme 1
  7. Security analysis of our scheme
  8. A novel asymptotically ideal CHSS scheme
  9. Our scheme 2
  10. Security analysis of our scheme
  11. Conclusion

Key Result

Lemma 1

Let $m_1,m_2,\ldots,m_n$ be positive integers that are pairwise coprime. Given any integers $x_{1},x_{2},\ldots,x_{n}$ and a system of congruences it holds that where $M=\prod\limits_{i=1}^{n}m_i$, $M_i=\frac{M}{m_i}$, and $\lambda_i\equiv M_i^{-1}\pmod {m_i}$. If the integer $x$ satisfies $0 \leq x < M$, the solution is unique, denoted by

Theorems & Definitions (28)

  • Lemma 1: CRT for integer ring, CRTdefinition
  • Definition 1: $k$-compact sequences of co-primes Tiplea2021
  • Remark 1
  • Remark 2
  • Definition 2: Secret sharing scheme
  • Definition 3: Information rate, Ning2018
  • Definition 4: Asymptotically ideal hierarchical secret sharing scheme, Quisquater2002
  • Definition 5: Asmuth-Bloom's secret sharing scheme, Asmuth-Bloom1983
  • Remark 3
  • Definition 6: Disjunctive hierarchical secret sharing scheme, Simmons1988
  • ...and 18 more