Learning with Rounding, Revisited - New Reduction, Properties and Applications

The learning with rounding (LWR) problem, introduced by Banerjee, Peikert and Rosen [BPR12] at EUROCRYPT ’12, is a variant of learning with errors (LWE), where one replaces random errors with deterministic rounding. The LWR problem was shown to be as hard as LWE for a setting of parameters where the modulus and modulus-to-error ratio are super-polynomial. In this work we resolve the main open problem of [BPR12] and give a new reduction that works for a larger range of parameters, allowing for a polynomial modulus and modulus-to-error ratio. In particular, a smaller modulus gives us greater efficiency, and a smaller modulus-to-error ratio gives us greater security, which now follows from the worst-case hardness of GapSVP with polynomial (rather than super-polynomial) approximation factors. As a tool in the reduction, we show that there is a “lossy mode” for the LWR problem, in which LWR samples only reveal partial information about the secret. This property gives us several interesting new applications, including a proof that LWR remains secure with weakly random secrets of sufficient minentropy, and very simple new constructions of deterministic encryption, lossy trapdoor functions and reusable extractors. Our approach is inspired by a technique of Goldwasser et al. [GKPV10] from ICS ’10, which implicitly showed the existence of a “lossy mode” for LWE. By refining this technique, we also improve on the parameters of that work to only requiring a polynomial (instead of super-polynomial) modulus and modulus-to-error ratio.