Batch Bootstrapping II:: Bootstrapping in Polynomial Modulus only Requires O~(1) FHE Multiplications in Amortization

This work continues the exploration of the batch framework proposed in Batch Bootstrapping I (Liu and Wang, Eurocrypt 2023). By further designing novel batch homomorphic algorithms based on the batch framework, this work shows how to bootstrap λ LWE input ciphertexts within a polynomial modulus, using O ~ ( λ ) FHE multiplications. This implies an amortized complexity O ~ ( 1 ) FHE multiplications per input ciphertext, significantly improving our first work (whose amortized complexity is O ~ ( λ 0.75 ) ) and the theoretical state of the art MS18 (Micciancio and Sorrell, ICALP 2018), whose amortized complexity is O ( 3 1 / ϵ · λ ϵ ) , for any arbitrary constant ϵ . We believe that all our new homomorphic algorithms might be useful in general applications, and thus can be of independent interests.