Paper 2011/139

Shortest Lattice Vectors in the Presence of Gaps

Mingjie Liu, Xiaoyun Wang, Guangwu Xu, and Xuexin Zheng

Abstract

Given a lattice ${\mathcal L}$ with the $i$-th successive minimum $\lambda_i$, its $i$-th gap $\frac{\lambda_i}{\lambda_1}$ often provides useful information for analyzing the security of cryptographic scheme related to ${\mathcal L}$. This paper concerns short vectors for lattices with gaps. In the first part, a $\lambda_2$-gap estimation of LWE lattices with cryptographic significance is given. For some $\gamma'$, a better reduction from $BDD_{\gamma'}$ to $uSVP_{\gamma}$ is obtained in the presence of larger $\lambda_2$-gap. The second part of the paper shows that gaps among the successive minima lead to a more efficient SVP search algorithm. As far as we know, it is the first SVP algorithm exploiting lattices with gaps.