Paper 2025/488

Exploring General Cyclotomic Rings in Torus-Based Fully Homomorphic Encryption

Abstract

In this article, we develop algebraic tools for fully homomorphic encryption over the torus in the setting of general composite cyclotomic indices. Working in cyclotomic rings and fields beyond the power-of-two case, we reframe and optimize key primitives—reduction modulo $\Phi_M$, homomorphic evaluation of trace operators, blind extraction and the blind rotation used in bootstrapping—using systematic duality and trace techniques. Our approach yields a simpler, more modular description of bootstrapping, including a new systematic treatment of so-called “nega-cyclicity” conditions and featuring an optimal reduction of the input noise, and provides sharp error bounds showing that bootstrap noise growth remains mild compared with the classical power-of-two instantiation. In addition, we introduce a new fast packing strategy. These results broaden the algebraic toolkit for torus-based FHE and pave the way for new cryptographic constructions and applications.