Exploration of polynomial multiplication algorithms for homomorphic encryption schemes

Homomorphic encryption schemes allow performing computations in the ciphertext domain, without the need of the secret key. In most promising schemes based on the ring-learning with errors (R-LWE) problem, polynomial multiplication operation is considered an important bottleneck. In this study, a comparison between the Karatsuba and the fast Fourier transform (FFT) multiplication algorithms in the context of homomorphic encryption is proposed in terms of complexity, flexibility and possible optimizations. A complete hardware architecture to speed up polynomial multiplication is provided and impacts of such an architecture on the Karatsuba and the FFT algorithms is thoroughly studied. The study demonstrates that in a realistic architecture, Karatsuba can be a better alternative than the FFT one.