# Statistical Decoding

## TL;DR

This paper analyzes statistical decoding, a randomized approach for code-based cryptography, providing its asymptotic complexity, efficient computation methods, and bounds, showing it cannot outperform Prange's algorithm at the Gilbert-Varshamov bound.

## Contribution

It offers the first detailed complexity analysis of statistical decoding, introduces efficient computation techniques, and establishes lower bounds on its performance.

## Key findings

- Provides asymptotic complexity of statistical decoding.
- Develops efficient methods for computing parity-check equations.
- Shows statistical decoding cannot outperform Prange's algorithm at the Gilbert-Varshamov bound.

## Abstract

The security of code-based cryptography relies primarily on the hardness of generic decoding with linear codes. The best generic decoding algorithms are all improvements of an old algorithm due to Prange: they are known under the name of information set decoding techniques (ISD). A while ago a generic decoding algorithm which does not belong to this family was proposed: statistical decoding. It is a randomized algorithm that requires the computation of a large set of parity-check equations of moderate weight. We solve here several open problems related to this decoding algorithm. We give in particular the asymptotic complexity of this algorithm, give a rather efficient way of computing the parity-check equations needed for it inspired by ISD techniques and give a lower bound on its complexity showing that when it comes to decoding on the Gilbert-Varshamov bound it can never be better than Prange's algorithm.