An improved BKW algorithm for LWE with applications to cryptography and lattices.

*(English)*Zbl 1336.94058
Gennaro, Rosario (ed.) et al., Advances in cryptology – CRYPTO 2015. 35th annual cryptology conference, Santa Barbara, CA, USA, August 16–20, 2015. Proceedings. Part I. Berlin: Springer (ISBN 978-3-662-47988-9/pbk; 978-3-662-47989-6/ebook). Lecture Notes in Computer Science 9215, 43-62 (2015).

Summary: In this paper, we study the Learning With Errors problem and its binary variant, where secrets and errors are binary or taken in a small interval. We introduce a new variant of the Blum, Kalai and Wasserman algorithm, relying on a quantization step that generalizes and fine-tunes modulus switching. In general this new technique yields a significant gain in the constant in front of the exponent in the overall complexity. We illustrate this by solving within half a day a \(\mathsf {LWE}\) instance with dimension \(n=128\), modulus \(q=n^{2}\), Gaussian noise \({\alpha} =1/(\sqrt{n/{\pi}} \log ^2 n)\) and binary secret, using \(2^{28}\) samples, while the previous best result based on BKW claims a time complexity of \(2^{74}\) with \(2^{60}\) samples for the same parameters.{

}We then introduce variants of \(\mathsf {BDD}\), \(\mathsf {GapSVP}\) and \(\mathsf {UniqueSVP}\), where the target point is required to lie in the fundamental parallelepiped, and show how the previous algorithm is able to solve these variants in subexponential time. Moreover, we also show how the previous algorithm can be used to solve the \(\mathsf {BinaryLWE}\) problem with \(n\) samples in subexponential time \(2^{(\ln 2/2+o(1))n/\log \log n}\). This analysis does not require any heuristic assumption, contrary to other algebraic approaches; instead, it uses a variant of an idea by Lyubashevsky to generate many samples from a small number of samples. This makes it possible to asymptotically and heuristically break the \(\mathsf {NTRU}\) cryptosystem in subexponential time (without contradicting its security assumption). We are also able to solve subset sum problems in subexponential time for density \(o\)(1), which is of independent interest: for such density, the previous best algorithm requires exponential time. As a direct application, we can solve in subexponential time the parameters of a cryptosystem based on this problem proposed at TCC 2010 [V. Lyubashevsky et al., Lect. Notes Comput. Sci. 5978, 382–400 (2010; Zbl 1274.94096)].

For the entire collection see [Zbl 1319.94002].

}We then introduce variants of \(\mathsf {BDD}\), \(\mathsf {GapSVP}\) and \(\mathsf {UniqueSVP}\), where the target point is required to lie in the fundamental parallelepiped, and show how the previous algorithm is able to solve these variants in subexponential time. Moreover, we also show how the previous algorithm can be used to solve the \(\mathsf {BinaryLWE}\) problem with \(n\) samples in subexponential time \(2^{(\ln 2/2+o(1))n/\log \log n}\). This analysis does not require any heuristic assumption, contrary to other algebraic approaches; instead, it uses a variant of an idea by Lyubashevsky to generate many samples from a small number of samples. This makes it possible to asymptotically and heuristically break the \(\mathsf {NTRU}\) cryptosystem in subexponential time (without contradicting its security assumption). We are also able to solve subset sum problems in subexponential time for density \(o\)(1), which is of independent interest: for such density, the previous best algorithm requires exponential time. As a direct application, we can solve in subexponential time the parameters of a cryptosystem based on this problem proposed at TCC 2010 [V. Lyubashevsky et al., Lect. Notes Comput. Sci. 5978, 382–400 (2010; Zbl 1274.94096)].

For the entire collection see [Zbl 1319.94002].

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\textit{P. Kirchner} and \textit{P.-A. Fouque}, Lect. Notes Comput. Sci. 9215, 43--62 (2015; Zbl 1336.94058)

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