Improved algorithms for the Shortest Vector Problem and the Closest Vector Problem in the infinity norm

TL;DR

This paper introduces a new linear-time sieving algorithm for the Shortest Vector Problem in the infinity norm, significantly improving efficiency and extending to approximate CVP, with analysis of heuristic algorithms in this norm.

Contribution

The paper presents a linear-time sieving procedure for SVP in the infinity norm and extends it to approximate CVP, improving upon previous quadratic-time algorithms.

Findings

Linear-time sieving algorithm for SVP in $oldsymbol{\ell_ extbf{infty} extbf{}}$ norm.

Faster algorithms for approximate SVP and CVP in the infinity norm.

Analysis of heuristic sieving algorithms in the $oldsymbol{\ell_ extbf{infty} extbf{}}$ norm, including volume intersection calculations.

Abstract

Blomer and Naewe[BN09] modified the randomized sieving algorithm of Ajtai, Kumar and Sivakumar[AKS01] to solve the shortest vector problem (SVP). The algorithm starts with $N = 2^{O(n)}$ randomly chosen vectors in the lattice and employs a sieving procedure to iteratively obtain shorter vectors in the lattice. The running time of the sieving procedure is quadratic in $N$. We study this problem for the special but important case of the $\ell_\infty$ norm. We give a new sieving procedure that runs in time linear in $N$, thereby significantly improving the running time of the algorithm for SVP in the $\ell_\infty$ norm. As in [AKS02,BN09], we also extend this algorithm to obtain significantly faster algorithms for approximate versions of the shortest vector problem and the closest vector problem (CVP) in the $\ell_\infty$ norm. We also show that the heuristic sieving algorithms of Nguyen and Vidick[NV08] and Wang et al.[WLTB11] can also be analyzed in the $\ell_{\infty}$ norm. The main technical contribution in this part is to calculate the expected volume of intersection of a unit ball centred at origin and another ball of a different radius centred at a uniformly random point on the boundary of the unit ball. This might be of independent interest.