# A Note on Lower Digits Extraction Polynomial for Bootstrapping

## TL;DR

This paper simplifies the construction of low-degree polynomials for digit extraction in FHE bootstrapping, establishes their optimality, and explores the limits of extracting other digits, contributing to more efficient cryptographic schemes.

## Contribution

It offers a simpler polynomial construction matching previous bounds, proves its optimality, and investigates the feasibility of extracting other low-order digits.

## Key findings

- Simpler polynomial construction matching asymptotic bounds
- Proof of optimality and limits of the approach
- Negative results on extracting other low-order digits

## Abstract

Bootstrapping is a crucial but computationally expensive step for realizing Fully Homomorphic Encryption (FHE). Recently, Chen and Han (Eurocrypt 2018) introduced a family of low-degree polynomials to extract the lowest digit with respect to a certain congruence, which helps improve the bootstrapping for both FV and BGV schemes. In this note, we present the following relevant findings about the work of Chen and Han (referred to as CH18): 1. We provide a simpler construction of the low-degree polynomials that serve the same purpose and match the asymptotic bound achieved in CH18; 2. We show the optimality and limit of our approach by solving a minimal polynomial degree problem; 3. We consider the problem of extracting other low-order digits using polynomials, and provide negative results.