Algorithms for Sparse LPN and LSPN Against Low-noise

TL;DR

This paper introduces a new polynomial-space algorithmic framework for sparse LPN and LSPN problems, improving efficiency and noise tolerance over previous methods in computational learning theory and cryptography.

Contribution

The authors develop a simple, polynomial-space framework that enhances learning algorithms for sparse LPN and LSPN, with improved time complexity and noise resilience.

Findings

LSPN algorithm runs in time O(η·n/k)^k for any noise rate η.

Sparse LPN learning algorithm improves over previous methods for certain noise levels.

Framework achieves polynomial space complexity, simplifying previous approaches.

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, this framework has a simple structure and runs in polynomial space. Let $n$ be the dimension, $k$ denote the sparsity, and $\eta$ be the noise rate. As a fundamental problem in computational learning theory, Learning Sparse Parities with Noise (LSPN) assumes the hidden parity is $k$-sparse. While a simple enumeration algorithm takes ${n \choose k}=O(n/k)^k$ time, previously known results stills need ${n \choose k/2} = \Omega(n/k)^{k/2}$ time for any noise rate $\eta$. 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 $\eta \in ( k/n,\sqrt{k/n})$. The sparse LPN problem is closely related to the classical problem of refuting random $k$-CSP and has been widely used in cryptography as the hardness assumption. Different from the standard LPN, 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 $e^{O(\eta \cdot n^{\frac{\delta+1}{2}})}$ time given $\delta \in (0,1)$ and $m \approx n^{1+(1-\delta)\cdot \frac{k-1}{2}}$ samples. This improves previous learning algorithms. For example, in the classical setting of $k=3$ and $m=n^{1.4}$, our algorithm would be faster than than previous approaches for any $\eta<n^{-0.7}$.