Robustness in Chinese Remainder Theorem

TL;DR

This paper introduces a new geometric approach to enhance the robustness of the Chinese Remainder Theorem, including a novel pseudo metric and a solution for multiple number reconstruction, addressing longstanding open problems.

Contribution

It proposes a pseudo metric for analyzing error bounds and dynamic range trade-offs, and presents the first robust CRT method for multiple numbers with polynomial-time solutions.

Findings

Introduced a pseudo metric for robust CRT analysis.

Developed a polynomial-time robust CRT for multiple numbers.

Provided theoretical insights into error bounds and dynamic range trade-offs.

Abstract

Chinese Remainder Theorem (CRT) has been widely studied with its applications in frequency estimation, phase unwrapping, coding theory and distributed data storage. Since traditional CRT is greatly sensitive to the errors in residues due to noises, the problem of robustly reconstructing integers via the erroneous residues has been intensively studied in the literature. In order to robustly reconstruct integers, there are two kinds of traditional methods: the one is to introduce common divisors in the moduli and the other is to directly decrease the dynamic range. In this paper, we take further insight into the geometry property of the linear space associated with CRT. Echoing both ways to introduce redundancy, we propose a pseudo metric to analyze the trade-off between the error bound and the dynamic range for robust CRT in general. Furthermore, we present the first robust CRT for multiple numbers to solve the problem of the CRT-based undersampling frequency estimation in general cases. Based on symmetric polynomials, we proved that in most cases, the problem can be solved in polynomial time efficiently. The work in this paper is towards a complete theoretical solution to the open problem over 20 years.