A Generalized Multidimensional Chinese Remainder Theorem (MD-CRT) for Multiple Integer Vectors

TL;DR

This paper extends the Chinese Remainder Theorem to multiple integer vectors with matrix moduli, addressing unique reconstruction and dynamic range conditions, with potential applications in multidimensional signal processing.

Contribution

It introduces a generalized MD-CRT for multiple vectors, deriving conditions for unique reconstruction and maximal dynamic range, improving upon existing scalar CRT results.

Findings

Derived a uniquely determinable range without prior info.

Proposed an algorithm to achieve the determinable range.

Established a new condition for maximal dynamic range in special cases.

Abstract

Chinese remainder theorem (CRT) is widely applied in cryptography, coding theory, and signal processing. It has been extended to the multidimensional CRT (MD-CRT), which reconstructs an integer vector from its vector remainders modulo multiple integer matrices. This paper investigates a generalized MD-CRT for multiple integer vectors, where the goal is to determine multiple integer vectors from multiple vector residue sets modulo multiple integer matrices.Comparing to the existing generalized CRT for multiple scalar integers, the challenge is that the moduli in MD-CRT are matrices that do not commute and the corresponding uniquely determinable range is multidimensional and the inclusion relationship is much more complicated. In this paper,we address two fundamental questions regarding the generalized MD-CRT. The first question concerns the uniquely determinable range of multiple integer vectors when no prior information about them is available. The second question is about the conditions under which the maximal possible dynamic range can be achieved.To answer these two questions, we first derive a uniquely determinable range without prior information and accordingly propose an algorithm to achieve it. A special case involving only two integer vectors is investigated for the second question, leading to a new condition for achieving the maximal possible dynamic range. Interestingly, this newly obtained condition, when the dimension is reduced to $1$, is even better than the existing ones for the conventional generalized CRT for scalar integers.These results may have applications for frequency detection in multidimensional signal processing.