Quantum Reduction of Finding Short Code Vectors to the Decoding Problem

TL;DR

This paper presents the first quantum reduction from finding short codewords in random linear codes to decoding in the Hamming metric, adapting lattice reduction techniques with new ingredients for the coarser metric.

Contribution

It introduces a novel quantum reduction for codes, extending lattice reduction ideas to the Hamming metric with new probabilistic and decoding radius techniques.

Findings

First quantum reduction from codeword finding to decoding

Adaptation of lattice reduction techniques to Hamming metric

Use of truncated Bernoulli distribution for errors

Abstract

We give a quantum reduction from finding short codewords in a random linear code to decoding for the Hamming metric. This is the first time such a reduction (classical or quantum) has been obtained. Our reduction adapts to linear codes Stehl\'e-Steinfield-Tanaka-Xagawa' re-interpretation of Regev's quantum reduction from finding short lattice vectors to solving the Closest Vector Problem. The Hamming metric is a much coarser metric than the Euclidean metric and this adaptation has needed several new ingredients to make it work. For instance, in order to have a meaningful reduction it is necessary in the Hamming metric to choose a very large decoding radius and this needs in many cases to go beyond the radius where decoding is always unique. Another crucial step for the analysis of the reduction is the choice of the errors that are being fed to the decoding algorithm. For lattices, errors are usually sampled according to a Gaussian distribution. However, it turns out that the Bernoulli distribution (the analogue for codes of the Gaussian) is too much spread out and cannot be used, as such, for the reduction with codes. This problem was solved by using instead a truncated Bernoulli distribution.