Hardness Amplification for (Sparse) LPN

• URL: http://arxiv.org/abs/2605.10056v2
• Date: Tue, 12 May 2026 03:03:30 GMT
• Title: Hardness Amplification for (Sparse) LPN
• Abstract summary: We prove new hardness amplification results for Learning Parity with Noise ($mathsfLPN$) and its sparse variants.<n>An algorithm that solves $mathsfLPN$ on a'small fraction of instances' can be converted into an algorithm that solves $mathsfLPN$ on 'almost all instances'<n>We extend the same amplification approach to $mathsfLPN$ over $mathbbF_q$ and to Sparse-$mathsfLPN$, where each query vector $vec a$ has
• Score: 5.1105538244022135
• License: http://creativecommons.org/licenses/by/4.0/
• 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 $\mathsf{LPN}$ 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.

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