Residue Number Systems (RNS) have been a topic of interest for years. Many previous works show that RNS is a good candidate for fast computations in asymmetric cryptography by using its intrinsic parallelization features. A recent result demonstrates that redundant RNS and modular reduction can fit together efficiently, providing an efficient RNS modular reduction algorithm owning a single-fault detection capability. In this paper, we propose to generalize this approach by protecting the classical Cox-Rower architecture against multi-fault attacks. We prove that faults occuring at different places and at different times can be detected with a linear cost for the architecture and a constant time for the execution.