Batch Bootstrapping I: - A New Framework for SIMD Bootstrapping in Polynomial Modulus.

In this series of work, we aim at improving the bootstrapping paradigm for fully homomorphic encryption (FHE). Our main goal is to show that the amortized cost of bootstrapping within a polynomial modulus only requires $$\tilde{O}(1)$$ FHE multiplications. To achieve this, we develop substantial algebraic techniques in two papers. Particularly, the first one (this work) proposes a new mathematical framework for batch homomorphic computation that is compatible with the existing bootstrapping methods of AP14/FHEW/TFHE. To show that our overall method requires only a polynomial modulus, we develop a critical algebraic analysis over noise growth, which might be of independent interest. Overall, the framework yields an amortized complexity $$\tilde{O}(\lambda ^{0.75})$$ FHE multiplications, where $$\lambda $$ is the security parameter. This improves the prior methods of AP14/FHEW/TFHE, which required $$O(\lambda )$$ FHE multiplications in amortization. Developing many substantial new techniques based on the foundation of this work, the sequel (Bootstrapping II, Eurocrypt 2023) shows how to further improve the recursive bootstrapping method of MS18 (Micciancio and Sorrell, ICALP 2018), yielding a substantial theoretical improvement that can potentially lead to more practical methods.