Cryptographic algorithm implementations are subject to specific attacks, called side channel attacks, focusing on the analysis of their power consumption or execution time or on the analysis of faulty computations. At FDTC06, Fumaroli and Vigilant presented a generic method to compute an exponentiation resistant against different side channel attacks. However, even if this algorithm does not reveal information on the secrets in case of a fault attack, it can not be used to safely implement a crypto-system involving an exponentiation. In this paper, we propose a new exponentiation method without this drawback and give a security proof of resistance to fault attacks. As an application, we propose an RSA algorithm implemented using the Chinese Remainder Theorem protected against side channel attacks. The exponentiation algorithm is also33% faster than the previous method.