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Novel CRT-based Asymptotically Ideal Disjunctive Hierarchical Secret Sharing Scheme

Hongju Li, Jian Ding, Fuyou Miao, Cheng Wang, Cheng Shu

Abstract

Disjunctive Hierarchical Secret Sharing (DHSS)} scheme is a type of secret sharing scheme in which the set of all participants is partitioned into disjoint subsets, and each subset is said to be a level with different degrees of trust and different thresholds. In this work, we focus on the Chinese Remainder Theorem (CRT)-based DHSS schemes due to their ability to accommodate flexible share sizes. We point out that the ideal DHSS scheme of Yang et al. (ISIT, 2024) and the asymptotically ideal DHSS scheme of Tiplea et al. (IET Information Security, 2021) are insecure. Consequently, existing CRT-based DHSS schemes either exhibit security flaws or have an information rate less than $\frac{1}{2}$. To address these limitations, we propose a CRT-based asymptotically perfect DHSS scheme that supports flexible share sizes. Notably, our scheme is asymptotically ideal when all shares are equal in size. Its information rate achieves one and it has computational security.

Novel CRT-based Asymptotically Ideal Disjunctive Hierarchical Secret Sharing Scheme

Abstract

Disjunctive Hierarchical Secret Sharing (DHSS)} scheme is a type of secret sharing scheme in which the set of all participants is partitioned into disjoint subsets, and each subset is said to be a level with different degrees of trust and different thresholds. In this work, we focus on the Chinese Remainder Theorem (CRT)-based DHSS schemes due to their ability to accommodate flexible share sizes. We point out that the ideal DHSS scheme of Yang et al. (ISIT, 2024) and the asymptotically ideal DHSS scheme of Tiplea et al. (IET Information Security, 2021) are insecure. Consequently, existing CRT-based DHSS schemes either exhibit security flaws or have an information rate less than . To address these limitations, we propose a CRT-based asymptotically perfect DHSS scheme that supports flexible share sizes. Notably, our scheme is asymptotically ideal when all shares are equal in size. Its information rate achieves one and it has computational security.
Paper Structure (11 sections, 7 theorems, 48 equations, 1 table)

This paper contains 11 sections, 7 theorems, 48 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Secret sharing
  4. Chinese Reminder Theorem and one-way hash function
  5. Security analysis of Yang et al. scheme
  6. Review of Yang et al. scheme
  7. An attack method on the Yang et al. scheme
  8. A novel asymptotically ideal DHSS scheme
  9. Our scheme
  10. Security analysis of our scheme
  11. Conclusion

Key Result

Lemma 1

Let $\mathbb{F}$ be a finite field and $m_1(x),m_2(x),\ldots,\\m_n(x)\in \mathbb{F}[x]$ be pairwise coprime polynomials. Denote by $M(x)=\prod\limits_{i=1}^{n}m_i(x)$, $M_i(x)=M(x)/m_i(x)$, and $\lambda_i(x)\equiv M_i^{-1}(x)\pmod {m_i(x)}$. For any given polynomials $y_1(x),y_2(x),\\\ldots,y_n(x)\i it holds that If the degree of $y(x)$ satisfies $\deg(y(x))<\deg(M(x))$, the solution is unique an

Theorems & Definitions (20)

  • Definition 1: Secret sharing scheme
  • Definition 2: Information rate, Ning2018
  • Definition 3: Disjunctive hierarchical secret sharing scheme, Tassa2007
  • Definition 4: Asymptotically ideal DHSS scheme, Quisquater2002
  • Lemma 1: CRT for polynomial ring, Ning2018Ding2023
  • Definition 5: One-way hash function, RM1985
  • Remark 1
  • Remark 2
  • Theorem 1: Correctness
  • proof
  • ...and 10 more