The Noisy Linear Equations problem involves finding the solution to a random linear system over a finite field given noisy evaluations. This is a generalisation of various learning problems widely used in cryptography, including Learning with Errors (LWE), Learning with Rounding (LWR), and Learning Parity with Noise (LPN).

We present a new sample-preserving search-to-decision reduction for Noisy Linear Equations that has complexity $\mathrm{poly}(n,m,\log q, q/\sigma)$, where $n$ is the dimension of the secret, $m$ is the number of samples, $q$ is the modulus, and $\sigma$ is the magnitude of the noise (defined suitably for the kind of noise involved). In particular, this implies search-to-decision reductions for LWE and LWR that run in time $\poly(n)$ even if the modulus $q$ is exponentially large, as long as the noise-to-modulus ratio $\sigma/q$ is non-negligible. Our reduction works for prime moduli $q$. If $q$ is composite, it instead produces a list that contains a small multiple of the solution.

Prior to this work, known search-to-decision reductions for LWE and LWR either did not preserve the parameters of the problem (i.e., $m$, $n$, $\sigma$, and $q$), or had complexity that scaled polynomially with either $\sigma$ or the largest prime factor of $q$.