State-of-the-art implementations of homomorphic encryption exploit the Fan and Vercauteren (FV) scheme and the Residue Number System (RNS). While the RNS breaks down large integer arithmetic into smaller independent channels, its non-positional nature makes operations such as division and rounding hard to implement, and makes the representation of small values suboptimal.

In this work, we propose the application of the Hybrid Position-Residues Number System representation to the FV scheme. This is a positional representation of large radix where the digits are represented in RNS. It inherits the benefits from RNS and allows to accelerate the critical division and rounding operations while also making the representation of smaller values more compact. This directly benefits the decryption and the homomorphic multiplication procedures of the FV scheme, reducing their asymptotic complexity and results in noticeable speedups when experimentally compared to related art RNS implementations.