Approximation Methods Part II: Approximations of Standard Functions

Chapter 5 described some commonly used general approximation methods and their trade-offs when used under fully homomorphic encryption (FHE). The reader is now ready to learn how to apply these and more targeted methods for estimating specific functions that are commonly used by modern analytics. We will describe how these functions can be estimated both in integer and approximate FHE contexts and then discuss the performance of the described methods in terms of time and accuracy. The chapter will start with a description of various methods for comparing encrypted values under FHE, as required when computing conditional code, sorting, or finding the maximum among a set of values under FHE. Section 6.2 builds upon these methods to compute the \({\operatorname {ReLU}} \) activation function used in neural networks (NNs). Section 6.3 describes how to compute reciprocals, both in the approximate and integer cases. Finally, Sect. 6.4 describes Chebyshev polynomials and how they can be used in the context of interpolation and function estimation, specifically focusing on trigonometric functions.