An Improved BKW Algorithm for LWE with Applications to Cryptography and Lattices

Abstract

Abstract. 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 LWE instance with dimension n = 128, modulus q = n 2 , Gaussian noise α = 1/( n/π 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 BDD, GapSVP and 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 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 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.

A Subfield Lattice Attack on Overstretched NTRU Assumptions

“…We verify that having such choice of super-linear n makes the Kirchner-Fouque [KF15] attack at least exponential in λ: exp(Θ(n/ log log q)) = exp(Θ (λlog ε λ log log λ/ log log 1+ε λ)) > exp(Θ(λ)).…”

“…In practice, the best known algorithm for attacking NTRU lattices is the combined lattice-reduction and meet-in-the-middle attack of Howgrave-Graham [HG07]. Asymptotically, a slightly sub-exponential attack against the ternary-NTRU problem was proposed by Kirchner and Fouque [KF15], with a heuristic complexity 2 Θ(n/ log log q) , which is to our knowledge the only sub-exponential attack when q is polynomial in n.…”

The Asymptotic Complexity of Coded-BKW with Sieving Using Increasing Reduction Factors

“…The problem with the uneven noise distribution was adressed independently in [11], [12]. The idea was to use a small step size and almost reduce the positions in the a vectors to 0 in the first step, and then gradually increase the step size n i and use less strict reduction for each step.…”

“…This was improved in [10] using Lazy Modulus Switching (LMS). Further improvements were made in [11], [12] by using a varying step size and a varying degree of reduction. In [13] coded-BKW with sieving was introduced, where lattice sieving techniques were used to improve the BKW algorithm.…”

Chosen-Ciphertext Security from Subset Sum

“…In case δ is o(1) or even as small as O(1/ log n), the problem is considered to be hard. The best classical algorithm for solving Subset Sum is due to [19], and takes sub-exponential time for solving instances with δ = o(1) and time 2…”