License Information

Document Description

This document describes how to evaluate an arbitrary smooth function on a ciphertext in CKKS using Chebyshev approximation. The Chebyshev approximation is a method of approximating a smooth function using polynomials. Note that Chebyshev evaluation only works with real inputs (or complex inputs with zero imaginary parts).

Example Description

The example for this code is located in function-evaluation.cpp. The file gives examples on how to run EvalLogistic, the logistic function $\frac{1}{1 + e^{-x}},$ and an arbitrary function using EvalChebyshevFunction. We use the square root function in our example for EvalChebyshevFunction.

Input Parameters

• ciphertext: This is the ciphertext we wish to operate on.
• a: This is the lower bound of underlying plaintext values we could have.
• b: This is the upper bound of underlying plaintext values we could have.
• degree: This is the polynomial degree of the Chebyshev approximation. A higher degree gives a more precise estimate, but takes longer to run.

How to Choose Multiplicative Depth

Each run of EvalChebyshevFunction requires a certain number of multiplications which depends on the input polynomial degree. We give a table below to map polynomial degrees to multiplicative depths.

Degree	Multiplicative Depth
3-5	4
6-13	5
14-27	6
28-59	7
60-119	8
120-247	9
248-495	10
496-1007	11
1008-2031	12
2032-4031	13
4032-8127	14
8128-16255	15
16256-32639	16
32640-65279	17
65280-130815	18
130816-261631	19

Note that if we use a range $(a, b) = (-1, 1),$ the multiplicative depth is 1 less than the depths listed in the table.