Parallel operations over TFHE-encrypted multi-digit integers

Recent advances in Fully Homomorphic Encryption (FHE) allow for a practical evaluation of non-trivial functions over encrypted data. In particular, novel approaches for combining ciphertexts broadened the scope of prospective applications. However, for arithmetic circuits, the overall complexity grows with the desired precision and there is only a limited space for parallelization. In this paper, we put forward several methods for fully parallel addition of multi-digit integers encrypted with the TFHE scheme. Since these methods handle integers in a special representation, we also revisit the signum function, firstly addressed by Bourse et al., and we propose a method for the maximum of two numbers; both with particular respect to parallelization. On top of that, we outline an approach for multiplication by a known integer. According to our experiments, the fastest approach for parallel addition of 31-bit encrypted integers in an idealized setting with 32 threads is estimated to be more than 6× faster than the fastest sequential approach. Finally, we demonstrate our algorithms on an evaluation of a practical neural network.