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Multi-bit Leveled Homomorphic Encryption via \(\mathsf {Dual.LWE}\)-Based

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Information Security and Cryptology (Inscrypt 2016)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 10143))

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Abstract

Fully Homomorphic Encryption (\(\mathsf {FHE}\)) is a cryptographic primitive that allows computing over encrypted data without decrypting the corresponding ciphertexts. In general, existing \(\mathsf {FHE}\) schemes can be achieved using standard Learning with Errors (\(\mathsf {LWE}\)) assumption and most of the schemes are single-bit encryption. Hence, the construction of multi-bit \(\mathsf {FHE}\) with high efficiency remains an open problem in cryptography. In this paper, we propose multi-bit versions of Public Key Encryption (\(\mathsf {PKE}\)) via the dual \(\mathsf {LWE}\)-based firstly proposed by Gentry, Peikert, and Vaikuntanathan at STOC 2008. We initially develop an universal construction derived from a general structure of the underlying combined public matrix for constructing the multi-bit version which increases the size of ciphertexts linearly. Then, utilizing multi-bit \(\mathsf {PKE}\) scheme as building block, we propose a new multi-bit \(\mathsf {FHE}\) scheme under the assumption of decisional \(\mathsf {LWE}\) is hard and prove the scheme is \(\textsc {IND-CPA}\)-secure.

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Notes

  1. 1.

    It is worth mentioning that \(\mathsf {LWE}\) is a generalization of “learning parities with noise” (LPN) which is the special case where \(q = 2\) and \(\chi \) is a Bernoulli distribution over \(\{0, 1\}\).

  2. 2.

    If we choose \(\mathsf {Dual.LWE}\)-based with binary secret, i.e. \(\mathbf {e}\leftarrow \{0,1\}^{m}\), implying \(||\mathbf {error}||\le \sqrt{m\cdot N}\cdot B_{\chi ^{m}}\).

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Acknowledgements

We would like to thank all anonymous reviewers for their helpful advice and comments. This work was supported by the National Natural Science Foundation of China (Grant No. 61472097), Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20132304110017).

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Authors and Affiliations

  1. College of Computer Science and Technology, Harbin Engineering University, Harbin, 15001, China

    Zengpeng Li, Chunguang Ma & Gang Du

  2. State Key Laboratory of Information Security, Institute of Information Engineering, Chinese Academy of Sciences, Beijing, 100093, China

    Zengpeng Li, Chunguang Ma & Gang Du

  3. Institute of Computing, University of Campinas, Campinas, 13083-050, Brazil

    Eduardo Morais

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  1. Zengpeng Li
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Correspondence to Chunguang Ma .

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Editors and Affiliations

  1. School of Science, Hangzhou Normal University, Hangzhou, China

    Kefei Chen

  2. Institute of Information Engineering, SKLOIS, Beijing, China

    Dongdai Lin

  3. Snapchat Inc., Columbia University, New York, New York, USA

    Moti Yung

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Li, Z., Ma, C., Morais, E., Du, G. (2017). Multi-bit Leveled Homomorphic Encryption via \(\mathsf {Dual.LWE}\)-Based. In: Chen, K., Lin, D., Yung, M. (eds) Information Security and Cryptology. Inscrypt 2016. Lecture Notes in Computer Science(), vol 10143. Springer, Cham. https://doi.org/10.1007/978-3-319-54705-3_14

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