On the Hardness of LWE with Binary Error: Revisiting the Hybrid Lattice-Reduction and Meet-in-the-Middle Attack

Abstract

The security of many cryptographic schemes has been based on special instances of the Learning with Errors (LWE) problem, e.g., Ring-LWE, LWE with binary secret, or LWE with ternary error. However, recent results show that some subclasses are weaker than expected. In this work we show that LWE with binary error, introduced by Micciancio and Peikert, is one such subclass. We achieve this by applying the Howgrave-Graham attack on NTRU, which is a combination of lattice techniques and a Meet-in-the-Middle approach, to this setting. We show that the attack outperforms all other currently existing algorithms for several natural parameter sets. For instance, for the parameter set $n=256$, $m=512$, $q=256$, this attack on LWE with binary error only requires $2^{85}$ operations, while the previously best attack requires $2^{117}$ operations. We additionally present a complete and improved analysis of the attack, using analytic techniques. Finally, based on the attack, we give concrete hardness estimations that can be used to select secure parameters for schemes based on LWE with binary error.