Cryptanalysis of a Class of Ideal Secret Sharing Schemes Based on the CRT for Polynomial Rings

2026-10-05

AI Summary

This study addresses a theoretical flaw in the “ideal” secret sharing scheme proposed by Yang et al. based on the Chinese Remainder Theorem (CRT) over polynomial rings, whose security analysis overlooked information leakage from publicly disclosed polynomials. By integrating CRT over polynomial rings, combinatorics, and cryptanalytic techniques, this work constructs three categories of attack strategies for unauthorized subsets, successfully achieving secret reconstruction. The findings confirm that the scheme is insecure under all non-degenerate parameters, thereby completely refuting its claimed idealness and revealing analogous vulnerabilities in the authors’ other hierarchical schemes. Ultimately, this research exposes fundamental errors in the theoretical foundations of the original construction and provides critical security warnings for the design of secret sharing schemes based on algebraic structures.

Abstract

Based on the CRT for polynomial rings, Yang, Zhu, Fu and Xia (ISIT 2026) proposed a compartmented secret sharing scheme for the compartmented access structure with lower bounds, claiming it can be ideal. We exhibit three families of unauthorized subsets that reconstruct the secret. The scheme is therefore insecure, except for a degenerate choice of parameters in which the only authorized subset is the whole participant set. The flaw in the security analysis is that it does not take all published polynomials into account. Similar flaws appear in the same first author's hierarchical schemes (ISIT 2024, 2026).

Problem

• Secret Sharing Schemes
• Cryptanalysis
• Chinese Remainder Theorem
• Polynomial Rings
• Compartmented Access Structure

Innovation

• Cryptanalysis
• Secret Sharing Schemes
• Chinese Remainder Theorem
• Polynomial Rings
• Compartmented Access Structure