The Learning Parity with Noise (LPN) problem underlines several classic cryptographic primitives. Researchers have endeavored to demonstrate the algorithmic difficulty of this problem by attempting to find a reduction from the decoding problem of linear codes, for which several hardness results exist. Earlier studies used code smoothing as a technical tool to achieve such reductions, showing that they are possible for codes with vanishing rate. This has left open the question of attaining a reduction with positive-rate codes. Addressing this case, we characterize the efficiency of the reduction in terms of the parameters of the decoding and LPN problems. As a conclusion, we isolate the parameter regimes for which a meaningful reduction is possible and the regimes for which its existence is unlikely.

翻译：带噪奇偶学习（LPN）问题是若干经典密码原语的理论基础。研究者一直试图通过寻找从线性码解码问题到该问题的归约来论证其算法难度，而线性码解码问题本身已有若干硬度结果。早期研究采用码平滑作为技术工具实现此类归约，证明该归约在码率趋零的编码方案中可行，但未解决在正码率编码方案中实现归约的问题。针对正码率情形，我们通过解码问题与LPN问题的参数刻画了归约效率。最终，我们分离出可实现有效归约的参数区域与归约存在可能性较低的参数区域。