Speeding Up Barrett and Montgomery Modular Multiplications

This paper proposes two improved modular multiplication algorithms based on Barrett and Montgomery modular reduction. The algorithms are simple and especially suitable for hardware implementations. Four large sets of moduli for which the proposed methods apply are given and analyzed from a security point of view. By considering state of art the attacks on public-key cryptosystems, we prove that the proposed sets are safe to use in practice for both elliptic curve cryptography and RSA cryptosystems. We propose a hardware architecture for the modular multiplier that is based on our methods. The results show that, concerning the speed, our proposed architecture outperforms the modular multiplier based on standard modular multiplication for more than 50 %. Additionally, our design consumes less area compared to the standard solutions. Furthermore, we adapt these algorithms for finite fields of characteristic 2.