In this paper, we study the Learning With Errors problem and its binary variant, wheresecrets 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

p within half a day a 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 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.