(Finite) field work: Choosing the best encoding of numbers for FHE computation

Abstract

Fully Homomorphic Encryption (FHE) schemes operate over finite fields while many use cases call for real numbers, requiring appropriate encoding of the data into the scheme’s plaintext space. However, the choice of encoding can tremendously impact the computational effort on the encrypted data. In this work, we investigate this question for applications that operate over integers and rational numbers using p-adic encoding and the extensions p’s Complement and Sign-Magnitude, based on three natural metrics: the number of finite field additions, multiplications, and multiplicative depth. Our results are partly constructive and partly negative: For the first two metrics, an optimal choice exists and we state it explicitly. However, for multiplicative depth the optimum does not exist globally, but we do show how to choose this best encoding depending on the use-case.