Limitations of the decoding-to-LPN reduction via code smoothing

TL;DR

This paper investigates the limitations of reducing the decoding problem of linear codes to the Learning Parity with Noise (LPN) problem, especially focusing on the challenges posed by positive-rate codes and the efficiency of such reductions.

Contribution

It characterizes the parameter regimes where decoding-to-LPN reductions are feasible and highlights the limitations in regimes with positive-rate codes.

Findings

Reductions are feasible only in certain parameter regimes.

Code smoothing techniques face limitations with positive-rate codes.

The paper delineates regimes where reductions are unlikely to exist.

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.