Limitations of the decoding-to-LPN reduction via code smoothing
Madhura Pathegama, Alexander Barg
TL;DR
This work critically analyzes the feasibility of decoding-to-LPN reductions via code smoothing for positive-rate linear codes. It formalizes the reduction in terms of a smoothing distribution and a total-variation error, and shows that meaningful reductions require a balanced combination of a non-negligible bias and a smoothing distance that does not decay too quickly. The authors prove an impossibility result for fast convergence to uniformity when the code rate is positive and the error weight grows linearly, while demonstrating that slower, controlled smoothing can enable reductions at the cost of decoder performance. They also provide a constructive slow-smoothing regime in which both the smoothing distance and the bias scale as inverse polynomials, making reductions feasible though less efficient. Overall, the paper delineates the limits of transferring worst-case decoding hardness to average-case LPN via smoothing for positive-rate codes and outlines directions for extending the results to nonbinary codes and lattice-based reductions.
Abstract
The Learning Parity with Noise (LPN) problem underlines several classic cryptographic primitives. Researchers have attempted to demonstrate the algorithmic hardness of this problem by finding reductions from the decoding problem of linear codes, for which several hardness results exist. Earlier studies used code smoothing as a tool to achieve reductions 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.