Hardness Amplification for (Sparse) LPN

2026-05-11

AI Summary

This work addresses the challenge of transforming algorithms for the Learning Parity with Noise (LPN) problem and its sparse variants—currently effective only on a limited fraction of instances—into ones that succeed with high probability on almost all instances. Building upon the direct product framework of Hirahara and Shimizu, the paper introduces a novel self-amplification mechanism for LPN-type problems by integrating instance amplification with parameter scaling. This approach applies uniformly to standard LPN, LPN over finite fields $\mathbb{F}_q$, and Sparse-LPN with fixed sparsity, achieving success probability $1 - \delta$ under typical parameter regimes such as $\eta/k$ and $n/k$. The result substantially strengthens the theoretical foundation of the average-case hardness assumption underlying LPN.

Abstract

We prove new hardness amplification results for Learning Parity with Noise ($\mathsf{LPN}$) and its sparse variants. In $\mathsf{LPN}_{η,n,m}$, 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 $e\leftarrow \mathrm{Ber}(η)$. Building on the direct-product framework introduced by Hirahara and Shimizu [HS23], we show an 'instance-fraction amplification' theorem: for any $\varepsilon,δ>0$, any algorithm that solves $\mathsf{LPN}_{η,n,m}$ with success probability $\varepsilon$ can be transformed into an algorithm that succeeds with probability $1-δ$ on a related \textsf{LPN} distribution with scaled parameters $\mathsf{LPN}_{η/k,\;n/k,\;m}$, where $ k=Θ\!\left(\frac{1}δ\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 $σ$ 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.

Problem

• LPN
• Hardness Amplification
• Sparse LPN
• Learning Parity with Noise
• Average-case Hardness

Innovation

• hardness amplification
• Learning Parity with Noise
• instance-fraction amplification
• Sparse-LPN
• self-amplification