# Faster provable sieving algorithms for the Shortest Vector Problem and the Closest Vector Problem on lattices in $\ell_p$ norm

## TL;DR

This paper introduces faster provable sieving algorithms for SVP and CVP on lattices in all $\, ext{l}_p$ norms, significantly improving time complexity over previous methods.

## Contribution

It presents a new linear sieving procedure applicable to all $\, ext{l}_p$ norms and a mixed sieving method that enhances efficiency, especially in $\, ext{l}_2$ norm.

## Key findings

- Achieves time complexity of $2^{2.751n+o(n)}$ for SVP and CVP.
- Improves $\, ext{l}_2$ norm sieving to $2^{2.25n+o(n)}$.
- Provides approximation algorithms with time complexity $2^{2.001n+o(n)}$.

## Abstract

In this work, we give provable sieving algorithms for the Shortest Vector Problem (SVP) and the Closest Vector Problem (CVP) on lattices in $\ell_p$ norm ($1\leq p\leq\infty$). The running time we obtain is better than existing provable sieving algorithms. We give a new linear sieving procedure that works for all $\ell_p$ norm ($1\leq p\leq\infty$). The main idea is to divide the space into hypercubes such that each vector can be mapped efficiently to a sub-region. We achieve a time complexity of $2^{2.751n+o(n)}$, which is much less than the $2^{3.849n+o(n)}$ complexity of the previous best algorithm. We also introduce a mixed sieving procedure, where a point is mapped to a hypercube within a ball and then a quadratic sieve is performed within each hypercube. This improves the running time, especially in the $\ell_2$ norm, where we achieve a time complexity of $2^{2.25n+o(n)}$, while the List Sieve Birthday algorithm has a running time of $2^{2.465n+o(n)}$. We adopt our sieving techniques to approximation algorithms for SVP and CVP in $\ell_p$ norm ($1\leq p\leq\infty$) and show that our algorithm has a running time of $2^{2.001n+o(n)}$, while previous algorithms have a time complexity of $2^{3.169n+o(n)}$.