Paper 2018/100

A Nonstandard Variant of Learning with Rounding with Polynomial Modulus and Unbounded Samples

Hart Montgomery

Abstract

The learning with rounding problem (LWR) has become a popular cryptographic assumption to study recently due to its determinism and resistance to known quantum attacks. Unfortunately, LWR is only known to be provably hard for instances of the problem where the LWR modulus $q$ is at least as large as some polynomial function of the number of samples given to an adversary, meaning LWR is provably hard only when (1) an adversary can only see a fixed, predetermined amount of samples or (2) the modulus $q$ is superpolynomial in the security parameter, meaning that the hardness reduction is from superpolynomial approximation factors on worst-case lattices. In this work, we show that there exists a (still fully deterministic) variant of the LWR problem that allows for both unbounded queries and a polynomial modulus $$, breaking an important theoretical barrier. To our knowledge, our new assumption, which we call the "nearby learning with lattice rounding problem" (NLWLR), is the first fully deterministic version of the learning with errors (LWE) problem that allows for both unbounded queries and a polynomial modulus. We note that our assumption is not practical for any kind of use and is mainly intended as a theoretical proof of concept to show that provably hard deterministic forms of LWE can exist with a modulus that does not grow polynomially with the number of samples.