A Simple Solution for Homomorphic Evaluation on Large Intervals

Homomorphic encryption (HE) is a promising technique used for privacy-preserving computation. Since HE schemes only support primitive polynomial operations, homomorphic evaluation of polynomial approximations for non-polynomial functions plays an important role in privacy-preserving machine learning. In this paper, we introduce a simple solution to approximating any functions, which might be overmissed by researchers: just using the neural networks for regressions. By searching decent superparameters, neural networks can achieve near-optimal computation depth for a given function with fixed precision, thereby reducing the modulus consumed. There are three main reasons why we choose neural networks for homomorphic evaluation of polynomial approximations. Firstly, neural networks with polynomial activation functions can be used to approximate whatever functions are needed in an encrypted state. This means that we can compute by one unified process for any polynomial approximation, such as that of Sigmoid or of ReLU. Secondly, by carefully finding an appropriate architecture, neural networks can efficiently evaluate a polynomial using near-optimal multiplicative depth, which would consume less modulus and therefore employ less ciphertext refreshing. Finally, as popular tools, model neural networks have many well-studied techniques that can conveniently serve our solution. Experiments showed that our method can be used for approximation of various functions. We exploit our method to the evaluation of the Sigmoid function on large intervals $[-30, +30]$, $[-50, +50]$, and $[-70, +70]$, respectively.