Abstract
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.
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Notes
- 1.
As this is the bottom base field, no further recursive acceleration can be applied (e.g., Karatsuba or Toom-Cook).
- 2.
The term non-trivial requires \(\textsf{Batch}\mathsf {\text {-}Mult}\) to be much more efficient than the trivial non-batch computation, i.e., computing r \(\textsf{RGSW}\) multiplications separately and then packing the outcomes into one ciphertext.
- 3.
Recall that the challenge is to homomorphically rotates batch ciphertexts of modes \(``\mathcal {R}_{12} \rightarrow \mathcal {R}_{13}"\) or \(``\mathcal {R}_{13} \rightarrow \mathcal {R}_{12}"\).
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Acknowledgement
The authors would like to thank anonymous reviewers for their insightful comments that significantly help improve the presentation. Feng-Hao Liu is supported by NSF CNS-1942400. Han Wang is supported by the National Key R &D Program of China under Grant 2020YFA0712303 and State Key Laboratory of Information Security under Grant TC20221013042.
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Liu, FH., Wang, H. (2023). Batch Bootstrapping II:. In: Hazay, C., Stam, M. (eds) Advances in Cryptology – EUROCRYPT 2023. EUROCRYPT 2023. Lecture Notes in Computer Science, vol 14006. Springer, Cham. https://doi.org/10.1007/978-3-031-30620-4_12
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