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The General Sieve Kernel and New Records in Lattice Reduction

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Advances in Cryptology – EUROCRYPT 2019 (EUROCRYPT 2019)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 11477))

Abstract

We propose the General Sieve Kernel (G6K, pronounced / e.si.ka/), an abstract stateful machine supporting a wide variety of lattice reduction strategies based on sieving algorithms. Using the basic instruction set of this abstract stateful machine, we first give concise formulations of previous sieving strategies from the literature and then propose new ones. We then also give a light variant of BKZ exploiting the features of our abstract stateful machine. This encapsulates several recent suggestions (Ducas at Eurocrypt 2018; Laarhoven and Mariano at PQCrypto 2018) to move beyond treating sieving as a blackbox SVP oracle and to utilise strong lattice reduction as preprocessing for sieving. Furthermore, we propose new tricks to minimise the sieving computation required for a given reduction quality with mechanisms such as recycling vectors between sieves, on-the-fly lifting and flexible insertions akin to Deep LLL and recent variants of Random Sampling Reduction.

Moreover, we provide a highly optimised, multi-threaded and tweakable implementation of this machine which we make open-source. We then illustrate the performance of this implementation of our sieving strategies by applying G6K to various lattice challenges. In particular, our approach allows us to solve previously unsolved instances of the Darmstadt SVP (151, 153, 155) and LWE (e.g. (75, 0.005)) challenges. Our solution for the SVP-151 challenge was found 400 times faster than the time reported for the SVP-150 challenge, the previous record. For exact-SVP, we observe a performance crossover between G6K and FPLLL’s state of the art implementation of enumeration at dimension 70.

The research of MA was supported by EPSRC grants EP/P009417/1, EP/S02087X/1 and by the European Union Horizon 2020 Research and Innovation Program Grant 780701; the research of LD was supported by a Veni Innovational Research Grant from NWO under project number 639.021.645 and by the European Union Horizon 2020 Research and Innovation Program Grant 780701; the research of EP was supported by the EPSRC and the UK government as part of the Centre for Doctoral Training in Cyber Security at Royal Holloway, University of London (EP/P009301/1); the research of GH and EK was supported by ERC Starting Grant ERC-2013-StG-335086-LATTAC.

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Notes

  1. 1.

    For example, the Gauss sieve implemented in FPLLL (latsieve) beats its unpruned SVP oracle (fplll -a svp) in dimension 50.

  2. 2.

    Our implementation is available at https://github.com/fplll/g6k/.

  3. 3.

    Note that, in addition, this already follows in the enumeration regime from [LN13] which we adapt to the sieving regime in Sect. 6.

  4. 4.

    Lifting is somewhat more expensive than considering a pair of vectors. We are therefore careful to only lift a fraction of all considered vectors, namely only the considered vectors below a certain length of, say, \(\sqrt{1.8} \cdot {{\,\mathrm{gh}\,}}(\mathcal {L}_{[\ell :r]})\).

  5. 5.

    The alternative being to only consider the vectors of the final database for lifting.

  6. 6.

    Note that \({{\,\mathrm{sl}\,}}\) can be viewed as the trivial insertion of the vector \(v_{\kappa } = (1, 0, \dots ,0)\).

  7. 7.

    When possible we prefer to sample by summing random pairs of vectors from the database.

  8. 8.

    This sequence refers to SubSieve \(^{+}(\mathcal {L}, f)\) with Sieve being progressive [Duc18a].

  9. 9.

    A collision is when a new vector \(\mathbf v \) to be inserted in the database equals \(\pm \mathbf v _2\) for some \(\mathbf v _2\) already present in the database.

  10. 10.

    For Fig. 4 we choose yet more opportunism and do not increase \(\beta \) to \(\beta '\).

  11. 11.

    This relies on the fact that we do not use recursive filtering in bgj1: the asymptotically optimal choice from [BGJ15] mandates choosing the buckets centres in a structured way, which is not compatible with choosing them as db elements.

  12. 12.

    This unorderedset is in fact split into many parts to eliminate most blocking locks during a multi-threaded sieve.

  13. 13.

    This mismatch with theory can be explained by various kinds of overheads, but mostly by the dimensions for free trick: as \(f=\varTheta (d/\log d)\) is quasilinear, the slope will only very slowly converge to the asymptotic prediction.

  14. 14.

    The number \(f=16+d/12\) of dimensions for free is only meant to be a local approximation, as we asymptotically expect \(f=\varTheta (d/\log d)\) even for O(1)-approx-SVP [Duc18a].

  15. 15.

    One could choose \(\kappa =0\) to be entirely sure not to miss the solution during the lifting phase, but this increases the cost of lifting. Instead, we can choose \(\kappa \) such that \(\sqrt{\kappa } \sigma < {{\,\mathrm{gh}\,}}(\mathcal {L}_{[d-\kappa :d]})\), with a small margin of, say, five dimensions.

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Acknowledgements

We thank Kenny Paterson for discussing a previous version of this draft. We also thank Pierre Karpman for running some of our experiments.

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Authors and Affiliations

  1. Information Security Group, Royal Holloway, University of London, Egham, UK

    Martin R. Albrecht & Eamonn W. Postlethwaite

  2. Cryptology Group, CWI, Amsterdam, The Netherlands

    Léo Ducas & Marc Stevens

  3. ENS Lyon, Laboratoire LIP, Lyon, France

    Gottfried Herold & Elena Kirshanova

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  1. Martin R. Albrecht
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Correspondence to Eamonn W. Postlethwaite .

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Editors and Affiliations

  1. Technion, Haifa, Israel

    Yuval Ishai

  2. COSIC Group, KU Leuven, Heverlee, Belgium

    Vincent Rijmen

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Albrecht, M.R., Ducas, L., Herold, G., Kirshanova, E., Postlethwaite, E.W., Stevens, M. (2019). The General Sieve Kernel and New Records in Lattice Reduction. In: Ishai, Y., Rijmen, V. (eds) Advances in Cryptology – EUROCRYPT 2019. EUROCRYPT 2019. Lecture Notes in Computer Science(), vol 11477. Springer, Cham. https://doi.org/10.1007/978-3-030-17656-3_25

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