Algorithms for Sparse LPN and LSPN Against Low-noise

Abstract

We consider sparse variants of the classical Learning Parities with random Noise (LPN) problem. Our main contribution is a new algorithmic framework that provides learning algorithms against low-noise for both Learning Sparse Parities (LSPN) problem and sparse LPN problem. Different from previous approaches for LSPN and sparse LPN (Grig11; valiant2012finding; KKK18; RRS17; GKM), this framework has a simple structure without fast matrix multiplication or tensor methods such that its algorithms are easy to implement and run in polynomial space. Let $n$ be the dimension, $k$ denote the sparsity, and $\eta$ be the noise rate such that each label gets flipped with probability $\eta$.

As a fundamental problem in computational learning theory (Feldman09), Learning Sparse Parities with Noise (LSPN) assumes the hidden parity is $k$-sparse instead of a potentially dense vector. While the simple enumeration algorithm takes ${n\choose k}=O(n/k)^{k}$ time, previously known results stills need at least ${n\choose k/2}=\Omega(n/k)^{k/2}$ time for any noise rate $\eta$ (Grig11; valiant2012finding; KKK18). Our framework provides a LSPN algorithm runs in time $O(\eta\cdot n/k)^{k}$ for any noise rate $\eta$, which improves the state-of-the-art of LSPN whenever $\eta\in(\sqrt{k/n},k/n)$.

The sparse LPN problem is closely related to the classical problem of refuting random $k$-CSP (FKO06; RRS17; GKM) and has been widely used in cryptography as the hardness assumption (e.g., Alekhnovich; ABW10; ADINZ18; DIJL_sparseLPN). Different from the standard LPN that samples random vectors in $\mathbf{F}_{2}^{n}$, it samples random $k$-sparse vectors. Because the number of $k$-sparse vectors is ${n\choose k}<n^{k}$, sparse LPN has learning algorithms in polynomial time when $m>n^{k/2}$. However, much less is known about learning algorithms for constant $k$ like 3 and $m<n^{k/2}$ samples, except the Gaussian elimination algorithm of time $e^{\eta n}$. Our framework provides a learning algorithm in $e^{\tilde{O}(\eta\cdot n^{\frac{\delta+1}{2}})}$ time given $\delta\in(0,1)$ and $m=\max\{1,\frac{\eta\cdot n^{\frac{\delta+1}{2}}}{k^{2}}\}\cdot n^{1+(1-\delta)\cdot\frac{k-1}{2}}$ samples. This improves previous learning algorithms in a wide range of parameters. For example, in the classical setting of $k=3$ and $m=n^{1.4}$ (FKO06; ABW10), our algorithm would be faster than $e^{\eta n}$ for any $\eta<n^{-0.6}$.

Since these two algorithms are based on one algorithmic framework, our conceptual contribution is a connection between sparse LPN and LSPN.