Batch Bootstrapping II: - Bootstrapping in Polynomial Modulus only Requires O~(1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{O}(1)$$\end{document} 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 $$\lambda $$ LWE input ciphertexts within a polynomial modulus, using $$\tilde{O}(\lambda )$$ FHE multiplications. This implies an amortized complexity $$\tilde{O}(1)$$ FHE multiplications per input ciphertext, significantly improving our first work (whose amortized complexity is $$\tilde{O}(\lambda ^{0.75})$$ ) and the theoretical state of the art MS18 (Micciancio and Sorrell, ICALP 2018), whose amortized complexity is $$O(3^{1/\epsilon } \cdot \lambda ^{\epsilon })$$ , for any arbitrary constant $$\epsilon $$ . We believe that all our new homomorphic algorithms might be useful in general applications, and thus can be of independent interests.