We conduct a systematic study of solving the learning parity with noise problem (LPN) using neural networks. Our main contribution is designing families of two-layer neural networks that practically outperform classical algorithms in high-noise, low-dimension regimes. We consider three settings where the numbers of LPN samples are abundant, very limited, and in between. In each setting we provide neural network models that solve LPN as fast as possible. For some settings we are also able to provide theories that explain the rationale of the design of our models. Comparing with the previous experiments of Esser, Kubler, and May (CRYPTO 2017), for dimension $n = 26$, noise rate $\tau = 0.498$, the ''Guess-then-Gaussian-elimination'' algorithm takes 3.12 days on 64 CPU cores, whereas our neural network algorithm takes 66 minutes on 8 GPUs. Our algorithm can also be plugged into the hybrid algorithms for solving middle or large dimension LPN instances.

翻译：我们系统地研究了利用神经网络求解带噪声线性奇偶性（LPN）问题。主要贡献在于设计了一系列双层神经网络族，在高噪声、低维度的实际场景中优于经典算法。针对LPN样本数量充足、极度有限以及介于两者之间的三种情况，我们分别提供了能够尽可能快速求解LPN的神经网络模型。在某些情形下，我们还提供了解释模型设计原理的理论依据。与Esser、Kubler和May（CRYPTO 2017）之前的实验相比，在维度n=26、噪声率τ=0.498时，"猜测-高斯消元"算法在64个CPU核心上需3.12天，而我们的神经网络算法在8个GPU上仅需66分钟。该算法还可嵌入混合算法中，用于求解中等或高维度的LPN实例。