Multi-Stage Robust Chinese Remainder Theorem

Abstract

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 <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$\tau$</tex></formula> if <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$\tau$</tex></formula> 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 <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$\tau$</tex></formula> 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.