Low-Latency Bootstrapping for CKKS using Roots of Unity

2026-07-29

AI Summary

Traditional bootstrapping methods in CKKS homomorphic encryption suffer from high computational latency and large multiplicative depth, which significantly hinder their efficiency and practicality. This work proposes the Sparse Roots of Unity (SPRU) bootstrapping algorithm, which for the first time directly embeds the modulo-q additive group into the space of complex roots of unity, enabling natively efficient bootstrapping within the CKKS framework. By leveraging sparse polynomial evaluation, SPRU substantially reduces the required multiplicative depth and supports smaller ring dimensions. Experimental results based on the OpenFHE library demonstrate that, in scenarios involving a small number of ciphertext slots, SPRU achieves up to a 5× reduction in latency compared to existing approaches.

Abstract

We introduce Sparse Roots of Unity (SPRU) bootstrapping, a new bootstrapping algorithm for the CKKS homomorphic encryption scheme for approximate arithmetic. The original CKKS bootstrapping method relies on homomorphically evaluating a polynomial that approximates modular reduction modulo q. In contrast, SPRU bootstrapping directly embeds the additive group modulo q into the complex roots of unity, which can be evaluated natively in the CKKS scheme. This approach significantly reduces the multiplicative depth required for bootstrapping, enabling the use of a smaller ring dimension and improving efficiency. In practice, using the OpenFHE C++ library, SPRU bootstrapping achieves up to a 5x reduction in latency when applied to ciphertexts with a small number of slots.

Problem

• CKKS
• bootstrapping
• low-latency
• homomorphic encryption
• roots of unity

Innovation

• CKKS
• bootstrapping
• roots of unity
• low-latency
• homomorphic encryption