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On the Integer Polynomial Learning with Errors Problem

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Public-Key Cryptography – PKC 2021 (PKC 2021)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 12710))

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Abstract

Several recent proposals of efficient public-key encryption are based on variants of the polynomial learning with errors problem (PLWE\(^f\)) in which the underlying polynomial ring \(\mathbb {Z}_q[x]/f\) is replaced with the (related) modular integer ring \(\mathbb {Z}_{f(q)}\); the corresponding problem is known as Integer Polynomial Learning with Errors (I-PLWE\(^f\)). Cryptosystems based on I-PLWE\(^f\) and its variants can exploit optimised big-integer arithmetic to achieve good practical performance, as exhibited by the ThreeBears cryptosystem. Unfortunately, the average-case hardness of I-PLWE\(^f\) and its relation to more established lattice problems have to date remained unclear.

We describe the first polynomial-time average-case reductions for the search variant of I-PLWE\(^f\), proving its computational equivalence with the search variant of its counterpart problem PLWE\(^f\). Our reductions apply to a large class of defining polynomials f. To obtain our results, we employ a careful adaptation of Rényi divergence analysis techniques to bound the impact of the integer ring arithmetic carries on the error distributions. As an application, we present a deterministic public-key cryptosystem over integer rings. Our cryptosystem, which resembles ThreeBears, enjoys one-way (OW-CPA) security provably based on the search variant of I-PLWE\(^f\).

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Notes

  1. 1.

    As commonly done, we also impose irreducibility of f in the problem definitions, to avoid weaknesses such as those pointed out in [BCF20].

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Acknowledgments

This work was supported in part by European Union Horizon 2020 Research and Innovation Program Grant 780701, Australian Research Council Discovery Project Grant DP180102199, and by BPI-France in the context of the national project RISQ (P141580).

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Authors and Affiliations

  1. ENS de Lyon, Laboratoire LIP (University of Lyon, CNRS, ENSL, INRIA, UCBL), Lyon, France

    Julien Devevey & Damien Stehlé

  2. Faculty of Information Technology, Monash University, Clayton, Australia

    Amin Sakzad & Ron Steinfeld

  3. Institut Universitaire de France, Paris, France

    Damien Stehlé

Authors
  1. Julien Devevey
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  2. Amin Sakzad
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  3. Damien Stehlé
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  4. Ron Steinfeld
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Correspondence to Julien Devevey .

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Editors and Affiliations

  1. Texas A&M University, College Station, TX, USA

    Juan A. Garay

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Devevey, J., Sakzad, A., Stehlé, D., Steinfeld, R. (2021). On the Integer Polynomial Learning with Errors Problem. In: Garay, J.A. (eds) Public-Key Cryptography – PKC 2021. PKC 2021. Lecture Notes in Computer Science(), vol 12710. Springer, Cham. https://doi.org/10.1007/978-3-030-75245-3_8

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  • DOI: https://doi.org/10.1007/978-3-030-75245-3_8

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