Hardness Amplification for (Sparse) 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.