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High-Speed Signatures from Standard Lattices

Part of the Lecture Notes in Computer Science book series (LNSC,volume 8895)

Abstract

At CT-RSA 2014 Bai and Galbraith proposed a lattice-based signature scheme optimized for short signatures and with a security reduction to hard standard lattice problems. In this work we first refine the security analysis of the original work and propose a new 128-bit secure parameter set chosen for software efficiency. Moreover, we increase the acceptance probability of the signing algorithm through an improved rejection condition on the secret keys. Our software implementation targeting Intel CPUs with AVX/AVX2 and ARM CPUs with NEON vector instructions shows that even though we do not rely on ideal lattices, we are able to achieve high performance. For this we optimize the matrix-vector operations and several other aspects of the scheme and finally compare our work with the state of the art.

Keywords

  • Signature scheme
  • Standard lattices
  • Vectorization
  • Ivy bridge

P. Schwabe—This work was supported by the German Research Foundation (DFG) through the DFG Research Training Group GRK 1817/1, by the German Federal Ministry of Economics and Technology through Grant 01ME12025 SecMobil), by the Netherlands Organisation for Scientific Research (NWO) through Veni 2013 project 13114, and by the German Federal Ministry of Education and Research (BMBF) through EC-SPRIDE. Permanent ID of this document: c5e2da3f0d05a056a5490a5c9b88baa9. Date: 2014-09-04.

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Fig. 1.
Fig. 2.

Notes

  1. 1.

    There exists sieving algorithms which can exploit the ideal structure, but the speed-up is of no significance [24, 36]. Some first ideas towards attacks with lower complexity were sketched by Bernstein in his blog [8].

  2. 2.

    Omitting costly Gaussian sampling was also the motivation for the design of the GLP signature [21].

  3. 3.

    A software implementation of a constant time discrete Gaussian sampler using the Cumulative Distribution Table (CDT) approach was recently proposed by Bos et al. [12]. However, even for the small standard deviation required for lattice-based encryption schemes, the constant time requirement leads to a significant overhead.

  4. 4.

    We note here that there was some vagueness in the parameter selection in the original work [6], also noticed later by the authors of the paper [5].

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Acknowledgment

We would like to thank Patrick Weiden, Rafael Misoczki, Shi Bai, and Steven Galbraith for useful discussions. We would further like to thank the anonymous reviewers for their suggestions and comments.

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A Decoding Attack

A Decoding Attack

An approach for solving LWE that has not been considered in the original work [6] is the decoding attack. It is inspired by the nearest plane algorithm proposed by Babai [4]. For a given lattice basis and a given target vector, it returns a lattice vector that is relatively close to the target vector. Hence, improving the quality of the lattice basis yields a vector that is closer to the target vector. Lindner and Peikert [26] proposed the nearest planes algorithm, a generalization of the former that returns more than one vector and thereby enhances the previous algorithm with a trade-off between its runtime and the probability of returning the actual closest vector within the set of obtained vectors.

There is a continuous correspondence between the success probability of this attack and the Hermite delta. We follow the approach proposed by Lindner and Peikert [26] to predict this success probability. In short, they show how one can use the Geometric Series Assumption (GSA) in order to predict the length of the Gram-Schmidt vectors of a reduced basis, and this estimation in turn serves to predict the success probability of the attack. Together with an estimation of the running time of nearest plane – the authors propose \(2^{-16}\) s – and the runtime estimation for basis reduction (see Eq. (2)), it is possible to predict the runtime and success probability of nearest planes.

Optimizing the trade-offs between the time spent on the attack and its success probability is not trivial, but simulations of the attack show that it is in most cases preferable to run multiple attacks with small success probabilities. This technique is called randomization and was investigated by Liu and Nguyen (see [27]), together with a further improvement called pruning. In comparison to the big improvement achieved with randomization, pruning leads only to a moderate speedup. The maximal speedup achieved in [27] is about \(2^6\), while randomization can reduce the cost by a factor of \(2^{32}\). Since it turned out that the decoding-attack is outperformed by other attacks by far (and pruning is furthermore very hard to analyze), we focused on the randomized version.

Briefly speaking, [26] provides the tools necessary to estimate the expected runtime of the attack for a given set of attack parameters, and [27] proposed to minimize the expected runtime (i.e. the time for one attack divided by the success probability of the attack). We applied this technique to our instance (cf. Table 2).

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Dagdelen, Ö. et al. (2015). High-Speed Signatures from Standard Lattices. In: Aranha, D., Menezes, A. (eds) Progress in Cryptology - LATINCRYPT 2014. LATINCRYPT 2014. Lecture Notes in Computer Science(), vol 8895. Springer, Cham. https://doi.org/10.1007/978-3-319-16295-9_5

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