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.

翻译：我们证明了针对含噪奇偶性学习问题（$\mathsf{LPN}$）及其稀疏变体的全新困难性放大结果。在$\mathsf{LPN}_{η,n,m}$中，目标是从$m$个含噪线性样本$(\vec a,b)$中恢复秘密向量$\vec s\in\mathbb{F}_2^n$，其中$\vec a\leftarrow \mathbb{F}_2^n$服从均匀分布，$b=\langle \vec a,\vec s\rangle + e$且$e\leftarrow \mathrm{Ber}(η)$。基于Hirahara与Shimizu [HS23]引入的直积框架，我们证明了“实例比例放大”定理：对任意$\varepsilon,δ>0$，任何以成功概率$\varepsilon$求解$\mathsf{LPN}_{η,n,m}$的算法，均可转化为一个成功概率为$1-δ$的算法，该算法作用于参数缩放后的相关$\mathsf{LPN}$分布$\mathsf{LPN}_{η/k,\;n/k,\;m}$，其中$ k=Θ\!\left(\frac{1}δ\log\frac{1}{\varepsilon}\right)$。等价地，求解“小比例实例”的$\mathsf{LPN}$算法可转化为求解“几乎全部实例”的$\mathsf{LPN}$算法，从而在广泛参数范围内实现自放大。我们将相同的放大方法推广至$\mathbb{F}_q$上的$\mathsf{LPN}$以及稀疏$\mathsf{LPN}$（其中每个查询向量$\vec a$恰好包含$σ$个非零条目）。这些结果共同为一大类$\mathsf{LPN}$型问题建立了困难性自放大机制，强化了假设$\mathsf{LPN}$及其稀疏变体在平均情况下的困难性这一基础。