Bootstrapping as a Morphism: An Arithmetic Geometry Approach to Asymptotically Faster Homomorphic Encryption
Dongfang Zhao
TL;DR
This work treats FHE bootstrapping as a geometric projection rather than circuit evaluation. By modeling the ciphertext space as an affine scheme and isolating decryptable and fresh regions with the noise and fresh ideals, it introduces a morphism that realizes bootstrapping via an algebraic folding projection. The main result is a complete bootstrapping algorithm with complexity $O(d \cdot \text{poly}(\log q))$, effectively removing the decryption-depth dependence $L_{dec}$ from the runtime. This geometric framework offers a principled path toward practical, high-performance FHE and establishes a bridge between lattice cryptography and arithmetic geometry for bootstrapping design and analysis.
Abstract
Fully Homomorphic Encryption (FHE) provides a powerful paradigm for secure computation, but its practical adoption is severely hindered by the prohibitive computational cost of its bootstrapping procedure. The complexity of all current bootstrapping methods is fundamentally tied to the multiplicative depth of the decryption circuit, denoted $L_{dec}$, making it the primary performance bottleneck. This paper introduces a new approach to bootstrapping that completely bypasses the traditional circuit evaluation model. We apply the tools of modern arithmetic geometry to reframe the bootstrapping operation as a direct geometric projection. Our framework models the space of ciphertexts as an affine scheme and rigorously defines the loci of decryptable and fresh ciphertexts as distinct closed subschemes. The bootstrapping transformation is then realized as a morphism between these two spaces. Computationally, this projection is equivalent to solving a specific Closest Vector Problem (CVP) instance on a highly structured ideal lattice, which we show can be done efficiently using a technique we call algebraic folding. The primary result of our work is a complete and provably correct bootstrapping algorithm with a computational complexity of $O(d \cdot \text{poly}(\log q))$, where $d$ is the ring dimension and $q$ is the ciphertext modulus. The significance of this result lies in the complete elimination of the factor $L_{dec}$ from the complexity, representing a fundamental asymptotic improvement over the state of the art. This geometric perspective offers a new and promising pathway toward achieving truly practical and high-performance FHE.