# The Asymptotic Complexity of Coded-BKW with Sieving Using Increasing Reduction Factors

## TL;DR

This paper analyzes the asymptotic complexity of an improved coded-BKW algorithm with sieving for solving the LWE problem, achieving a slight complexity reduction through variable reduction factors.

## Contribution

It introduces a novel approach of using different reduction factors at each step of the sieving process in coded-BKW, leading to improved asymptotic complexity results.

## Key findings

- Asymptotic complexity improved to 2^{0.8917n} in the Regev setting
- Method achieves similar improvements under quantum computing assumptions
- Provides a detailed analysis of reduction factors in coded-BKW with sieving

## Abstract

The Learning with Errors problem (LWE) is one of the main candidates for post-quantum cryptography. At Asiacrypt 2017, coded-BKW with sieving, an algorithm combining the Blum-Kalai-Wasserman algorithm (BKW) with lattice sieving techniques, was proposed. In this paper, we improve that algorithm by using different reduction factors in different steps of the sieving part of the algorithm. In the Regev setting, where $q = n^2$ and $\sigma = n^{1.5}/(\sqrt{2\pi}\log_2^2 n)$, the asymptotic complexity is $2^{0.8917n}$, improving the previously best complexity of $2^{{0.8927n}}$. When a quantum computer is assumed or the number of samples is limited, we get a similar level of improvement.