This expository note shows that the learning parities with noise (LPN) assumption is robust to weak dependencies in the noise distribution of small batches of samples. This provides a partial converse to the linearization technique of [AG11].
The key insights are:
The main result, Theorem 1.4, shows that for any constant batch size k and any δ-Santha-Vazirani source p over the batch noise, the standard LPN problem with noise level 1/2 - O(kδ) is polynomial-time reducible to learning parities with the batch noise distribution p. This provides a robustness guarantee for the LPN assumption in the face of small dependencies in the noise.
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