A general approach for improving RNS Montgomery exponentiation using pre-processing

Filippo Gandino (1) , Fabrizio Lamberti (1) , Paolo Montuschi (1) , Jean-Claude Bajard (2)

1 DAUIN - Dipartimento di Automatica e Informatica [Torino]
2 PEQUAN - Performance et Qualité des Algorithmes Numériques

The hardware implementation of modular expo-nentiation for very large integers is a well-known topic in digital arithmetic. An effective approach for obtaining parallel and carry-free implementations consists in using the Mont-gomery exponentiation algorithm and executing the necessary operations in RNS. Two efficient methods for performing the RNS Montgomery exponentiation have been proposed by Kawamura et al. and by Bajard and Imbert. The above approaches mainly differ in the algorithm used for imple-menting the base extension. This paper presents a modified RNS Montgomery exponentiation algorithm, where several multiplications are moved outside the main execution loop and replaced by an effective pre-processing stage producing a significant saving on the overall delay with respect to state-of-the-art approaches. Since the proposed modification should be applied to both of the above algorithms, two versions are specifically discussed. The overall comparison shows that with the proposed approach, a 18.5% speedup can be achieved for an implementation over 1024 bits, without any significant area overhead.

Mots clés: -RNS; Montgomery reduction; Modular exponenti-ation; Modular multiplication

Domaines: Informatique [cs]

Filippo Gandino, Fabrizio Lamberti, Paolo Montuschi, Jean-Claude Bajard. A general approach for improving RNS Montgomery exponentiation using pre-processing. ARITH 2011 - 20th IEEE Symposium on Computer Arithmetic, Jul 2011, Tübingen, Germany. pp.195-204, ⟨10.1109/ARITH.2011.35⟩. ⟨hal-01098806⟩