Efficient RNS Bases for Cryptography

Residue Number Systems (RNS) are useful for distributing large dynamic range computations over small modular rings, which allows the speed up of computations. This feature is well known, and already used in both DSP and cryptography. In this paper we deal with implementation for huge numbers like those used for ciphering as with RSA or ECC on prime finite fields. Modular multiplication is the main operation of these protocols. We find very interesting modular multiplication algorithms in RNS where the conversion from an RNS basis to another represents the main part of the complexity. Hence, we propose in this paper an analysis of the criteria for selecting some bases giving efficient conversions. We conclude by giving methods for constructing an efficient basis in function of the size of different parameters like the basic operators, the key of the cryptosystem, etc. Residue Number Systems (RNS) are useful for distributing large dynamic range computations over small modular rings, which allows the speed up of computations. This feature is well known, and already used in both DSP and cryptography. In this paper we deal with implementation for huge numbers like those used for ciphering as with RSA or ECC on prime finite fields. Modular multiplication is the main operation of these protocols. We find very interesting modular multiplication algorithms in RNS where the conversion from an RNS basis to another represents the main part of the complexity. Hence, we propose in this paper an analysis of the criteria for selecting some bases giving efficient conversions. We conclude by giving methods for constructing an efficient basis in function of the size of different parameters like the basic operators, the key of the cryptosystem, etc.