Paper 2025/2164

Hardness and Algorithms for Batch LPN under Dependent Noise

Abstract

We study the Batch Learning Parity with Noise (LPN) variant, where the oracle returns $k$ samples in a batch and draws the noise vector from a joint noise distribution $\mathcal{D}$ over $\mathbb{F}_2^k$ instead of from an i.i.d. product distribution. This model captures a broad range of correlated or structured noise patterns studied in cryptography and learning theory. Consequently, understanding which dependent-noise distributions preserve the hardness of LPN has become an important question. On the hardness side, we design several reductions from standard LPN to Batch LPN. Our reductions identify broader classes of noise distributions $\mathcal{D}$ that preserve LPN hardness, extending the prior hardness results of Golowich, Moitra, and Rohatgi (FOCS 2024). We show hardness in three regimes: 1. If $\mathcal{D}$ satisfies a mild Fourier-analytic condition $\sum_{\mathbf{s}\neq \mathbf0}|\widehat{P}_{\mathcal{D}}(\mathbf{s})|\le 2\varepsilon$, then Batch LPN is as hard as standard LPN with noise rate $1/2-\varepsilon$; 2. If $\mathcal{D}$ is $\Omega(\eta \cdot k 2^{-k})$-dense (i.e., every error pattern occurs with probability at least $\Omega(\eta \cdot k 2^{-k})$) for $\eta < 1/k$, then Batch LPN is as hard as standard LPN with noise rate $\eta$; 3. If $\mathcal{D}$ is a $\delta$-Santha-Vazirani source, then Batch LPN is as hard as standard LPN with noise rate $1/2-\varepsilon$ whenever $\delta\le O(2^{-k/2}\varepsilon)$, improving the previous $O(2^{-k}\varepsilon)$ dependence (in Golowich et al.). On the algorithmic side, we extend Arora and Ge's (ICALP 2011) linearization attack to show that Batch LPN can be solved when the noise distribution assigns sufficiently small probability to at least one point, giving an algorithmic counterpart to our hardness results. Our reductions are based on random affine transformations and are analyzed through the lens of Fourier analysis, providing a general framework for studying dependent-noise LPN variants.