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Asymptotically Ideal Conjunctive Hierarchical Secret Sharing Scheme Based on CRT for Polynomial Ring

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

Abstract

Conjunctive Hierarchical Secret Sharing (CHSS) is a type of secret sharing that divides participants into multiple distinct hierarchical levels, with each level having a specific threshold. An authorized subset must simultaneously meet the threshold of all levels. Existing Chinese Remainder Theorem (CRT)-based CHSS schemes either have security vulnerabilities or have an information rate lower than $\frac{1}{2}$. In this work, we utilize the CRT for polynomial ring and one-way functions to construct an asymptotically perfect CHSS scheme. It has computational security, and permits flexible share sizes. Notably, when all shares are of equal size, our scheme is an asymptotically ideal CHSS scheme with an information rate one.

Asymptotically Ideal Conjunctive Hierarchical Secret Sharing Scheme Based on CRT for Polynomial Ring

Abstract

Conjunctive Hierarchical Secret Sharing (CHSS) is a type of secret sharing that divides participants into multiple distinct hierarchical levels, with each level having a specific threshold. An authorized subset must simultaneously meet the threshold of all levels. Existing Chinese Remainder Theorem (CRT)-based CHSS schemes either have security vulnerabilities or have an information rate lower than . In this work, we utilize the CRT for polynomial ring and one-way functions to construct an asymptotically perfect CHSS scheme. It has computational security, and permits flexible share sizes. Notably, when all shares are of equal size, our scheme is an asymptotically ideal CHSS scheme with an information rate one.
Paper Structure (8 sections, 7 theorems, 41 equations, 1 table)

This paper contains 8 sections, 7 theorems, 41 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. CRT for polynomial ring
  4. Secret sharing
  5. An asymptotically ideal CHSS scheme
  6. Our scheme
  7. Security analysis of our scheme
  8. Conclusion

Key Result

Lemma 1

Let $\mathbb{F}$ be a finite field, and let $m_1(x),m_2(x),\\\ldots,m_n(x)\in \mathbb{F}[x]$ be pairwise coprime polynomials. For any given polynomials $g_1(x),g_2(x),\\\ldots,g_n(x)\in \mathbb{F}[x]$, consider the system of congruences Then its solutions satisfies where Moreover, if the degree of $y(x)$ satisfies $\deg(y(x))<\deg(M(x))$, the solution is unique and can be written as

Theorems & Definitions (17)

  • Lemma 1: CRT for polynomial ring, Ning2018Ding2023
  • Definition 1: Secret sharing scheme
  • Definition 2: Information rate, Ning2018
  • Definition 3: Conjunctive hierarchical secret sharing scheme, Tassa2007
  • Definition 4: Asymptotically ideal CHSS scheme, Quisquater2002
  • Theorem 1: Correctness
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • ...and 7 more