Multi-Stage Robust Chinese Remainder Theorem

It is well-known that the traditional Chinese remainder theorem (CRT) is not robust in the sense that a small error in a remainder may cause a large reconstruction error. A robust CRT was recently proposed for a special case when the greatest common divisor (gcd) of all the moduli is more than 1 and the remaining integers factorized by the gcd are co-prime. It basically says that the reconstruction error is upper bounded by the remainder error level tau if tau is smaller than a quarter of the gcd of all the moduli. In this paper, we consider the robust reconstruction problem for a general set of moduli. We first present a necessary and sufficient condition on the remainder errors with a general set of moduli and also a corresponding robust reconstruction method. This can be thought of as a single-stage robust CRT. We then propose a two-stage robust CRT by grouping the moduli into several groups as follows. First, the single-stage robust CRT is applied to each group. Then, with these robust reconstructions from all the groups, the single-stage robust CRT is applied again across the groups. This is easily generalized to multi-stage robust CRT. With this two-stage robust CRT, the robust reconstruction holds even when the remainder error level tau is above the quarter of the gcd of all the moduli, and an algorithm on how to group a set of moduli for a better reconstruction robustness is proposed in some special cases.