Revisiting the Hardness of Binary Error LWE

Abstract

Binary error LWE is the particular case of the learning with errors (LWE) problem in which errors are chosen in $\{0,1\}$. It has various cryptographic applications, and in particular, has been used to construct efficient encryption schemes for use in constrained devices. Arora and Ge showed that the problem can be solved in polynomial time given a number of samples quadratic in the dimension n. On the other hand, the problem is known to be as hard as standard LWE given only slightly more than n samples.

In this paper, we first examine more generally how the hardness of the problem varies with the number of available samples. Under standard heuristics on the Arora–Ge polynomial system, we show that, for any $\epsilon >0$, binary error LWE can be solved in polynomial time $n^{O(1/\epsilon )}$ given $\epsilon \cdot n^{2}$ samples. Similarly, it can be solved in subexponential time $2^{\tilde{O}(n^{1-\alpha })}$ given $n^{1+\alpha }$ samples, for $0<\alpha <1$.

As a second contribution, we also generalize the binary error LWE to problem the case of a non-uniform error probability, and analyze the hardness of the non-uniform binary error LWE with respect to the error rate and the number of available samples. We show that, for any error rate $0< p < 1$, non-uniform binary error LWE is also as hard as worst-case lattice problems provided that the number of samples is suitably restricted. This is a generalization of Micciancio and Peikert’s hardness proof for uniform binary error LWE. Furthermore, we also discuss attacks on the problem when the number of available samples is linear but significantly larger than n, and show that for sufficiently low error rates, subexponential or even polynomial time attacks are possible.