A Multi-layer Recursive Residue Number System

Abstract

We present a method to increase the dynamical range of a Residue Number System (RNS) by adding virtual RNS layers on top of the original RNS, where the required modular arithmetic for a modulus on any non-bottom layer is implemented by means of an RNS Montgomery multiplication algorithm that uses the RNS on the layer below. As a result, the actual arithmetic is deferred to the bottom layer. The multiplication algorithm that we use is based on an algorithm by Bajard and Imbert, extended to work with pseudo-residues (remainders with a larger range than the modulus). The resulting Recursive Residue Number System (RRNS) can be used to implement modular addition, multiplication, and multiply-and-accumulate for very large (2000+ bits) moduli, using only modular operations for small (for example 8-bits) moduli. A hardware implementation of this method allows for massive parallelization.Our method can be applied in cryptographic algorithms such as RSA to realize modular exponentiation with a large (2048bit, or even 4096-bit) modulus. Due to the use of full RNS Montgomery algorithms, the system does not involve any carries, therefore cryptographic attacks that exploit carries cannot be applied.

Hierarchical Approach in RNS Base Extension for Asymmetric Cryptography

“…[18] and [19] propose to recursively use RNS where computations inside channels are performed in a small RNS base. This does not improve much performances compared to usual RNS, but it provides other properties such as protection against some physical attacks.…”

“…This does not improve much performances compared to usual RNS, but it provides other properties such as protection against some physical attacks. [19] proposes small RNS channels where all elementary operations are performed using fully precomputed lookup tables (reducing information leakage).…”