Résumé
Modular arithmetic in Residue Number System is generally done using Montogmery method for reduction. During the past decade, two attempts were proposed to perform modular reduction in RNS with Barrett method. The first one was due Schinianakis and Stouraisi in 2014, but they do not get their algorithm fully in RNS since it involves some multi-precision integer operations. The second one was due to Garg and Xiao in 2017, which was the first version of Barrett reduction method fully done in RNS. But both methods remain two costly compared to Montgomery approach in RNS. In this paper we present a novel version of Barrett multiplication and reduction which are fully done in RNS. The resulting complexities of these approaches improve the previous two versions of Barrett in RNS of the state of the art. Considering our approach for modular multiplication is remains more costly than multiplication Montgomery by t 2 multiplications and t 2 additions. But we get an improvement by t 2 additions and t 2 multiplication when doing only reduction. We provide implementation results for modular reduction and for matrix multiplication modulo large integer which confirm the improvement provided by the proposed approach.