Hardness Amplification for (Sparse) LPN

TL;DR

This paper proves new results on hardness amplification for Learning Parity with Noise (LPN) and its sparse variants, showing how algorithms solving LPN can be transformed to succeed on almost all instances, strengthening assumptions of average-case hardness.

Contribution

It introduces an instance-fraction amplification theorem for LPN, extending it to over _q and sparse variants, establishing broad hardness self-amplification results.

Findings

Transform algorithms with low success probability into high-success algorithms on most instances.

Extends amplification results to _q and sparse LPN variants.

Strengthens the theoretical foundation for the average-case hardness of LPN.

Abstract

We prove new hardness amplification results for Learning Parity with Noise ($\mathsf{LPN}$) and its sparse variants. In $\mathsf{LPN}_{\eta,n,m}$, the goal is to recover a secret $\vec s\in\mathbb{F}_2^n$ from $m$ noisy linear samples $(\vec a,b)$, where $\vec a\leftarrow \mathbb{F}_2^n$ is uniform and $b=\langle \vec a,\vec s\rangle + e$ with $e\leftarrow \mathrm{Ber}(\eta)$. Building on the direct-product framework introduced by Hirahara and Shimizu [HS23], we show an 'instance-fraction amplification' theorem: for any $\varepsilon,\delta>0$, any algorithm that solves $\mathsf{LPN}_{\eta,n,m}$ with success probability $\varepsilon$ can be transformed into an algorithm that succeeds with probability $1-\delta$ on a related $\mathsf{LPN}$ distribution with scaled parameters $\mathsf{LPN}_{\eta/k,\;n/k,\;m}$, where $ k=\Theta\!\left(\frac{1}{\delta}\log\frac{1}{\varepsilon}\right). $ Equivalently, an algorithm that solves $\mathsf{LPN}$ on a 'small fraction of instances' can be converted into an algorithm that solves $\mathsf{LPN}$ on 'almost all instances', yielding a self-amplification for a wide range of parameters. We extend the same amplification approach to $\mathsf{LPN}$ over $\mathbb{F}_q$ and to Sparse-$\mathsf{LPN}$, where each query vector $\vec a$ has exactly $\sigma$ nonzero entries. Together, these results establish hardness self-amplification for a broad family of $\mathsf{LPN}$-type problems, strengthening the foundations for assuming the average-case hardness of $\mathsf{LPN}$ and its sparse variants.