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Limits of Polynomial Packings for \(\mathbb {Z}_{p^k}\) and \(\mathbb {F}_{p^k}\)

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Advances in Cryptology – EUROCRYPT 2022 (EUROCRYPT 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13275))

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Abstract

We formally define polynomial packing methods and initiate a unified study of related concepts in various contexts of cryptography. This includes homomorphic encryption (HE) packing and reverse multiplication-friendly embedding (RMFE) in information-theoretically secure multi-party computation (MPC). We prove several upper bounds and impossibility results on packing methods for \(\mathbb {Z}_{p^k}\) or \(\mathbb {F}_{p^k}\)-messages into \(\mathbb {Z}_{p^t}[x]/f(x)\) in terms of (i) packing density, (ii) level-consistency, and (iii) surjectivity. These results have implications on recent development of HE-based MPC over \(\mathbb {Z}_{2^k}\) secure against actively corrupted majority and provide new proofs for upper bounds on RMFE.

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Notes

  1. 1.

    Chinese Remainder Theorem.

  2. 2.

    Single Instruction, Multiple Data.

  3. 3.

    The original method of [33] does not consider packings for \({\mathbb Z}_{p^k}\). Gentry-Halevi-Smart [24] later generalized the method to support such packing. However, this method achieves only considerably low efficiency. See Sect. 4.1.

  4. 4.

    Indeed, the number of evaluation points is bounded by the size of the field.

  5. 5.

    Zero-knowledge proof of message knowledge.

  6. 6.

    Nonetheless, this object was also previously studied in [2] to amortize oblivious linear evaluations (OLE) into a larger extension field for correlation extraction problem in MPC. However, their construction achieved only sublinear density.

  7. 7.

    Similar problems were also considered in other cryptography literature [2, 18, 29]. For more detailed discussions, see the full version [14].

  8. 8.

    In a sense that any element of \({\mathcal R}\) could be an image of \({\textsf {{Pack}}}_1(\cdot )\).

  9. 9.

    ZKPoMK was first conceptualized in MHZ2k [13], but it is also performed in Overdrive2k [31] implicitly. For detailed discussion, refer to [13].

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Acknowledgement

The authors thank Dongwoo Kim for insightful discussions on packing methods, Donggeon Yhee for discussions on Proposition 7, and Minki Hhan for constructive comments on an earlier version of this work. The authors also thank the reviewers of Eurocrypt 2022 who provided thoughtful suggestions to improve the earlier version of this paper. This work was supported by Institute of Information & Communications Technology Planning & Evaluation (IITP) grant funded by the Korea government (MSIT) (No. 2020-0-00840, Development and Library Implementation of Fully Homomorphic Machine Learning Algorithms supporting Neural Network Learning over Encrypted Data).

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Authors and Affiliations

  1. Seoul National University, Seoul, Republic of Korea

    Jung Hee Cheon & Keewoo Lee

  2. Crypto Lab Inc., Seoul, Republic of Korea

    Jung Hee Cheon

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  1. University of Haifa, Haifa, Haifa, Israel

    Orr Dunkelman

  2. University of Warsaw, Warsaw, Poland

    Stefan Dziembowski

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Cheon, J.H., Lee, K. (2022). Limits of Polynomial Packings for \(\mathbb {Z}_{p^k}\) and \(\mathbb {F}_{p^k}\). In: Dunkelman, O., Dziembowski, S. (eds) Advances in Cryptology – EUROCRYPT 2022. EUROCRYPT 2022. Lecture Notes in Computer Science, vol 13275. Springer, Cham. https://doi.org/10.1007/978-3-031-06944-4_18

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