Time-Memory Trade-Off for Lattice Enumeration in a Ball

Abstract

Enumeration algorithms in lattices are a well-known technique for solving the Short Vector Problem (SVP) and improving blockwise lattice reduction algorithms. Here, we propose a new algorithm for enumerating lattice point in a ball of radius $1.156\lambda_1(\Lambda)$ in time $3^{n+o(n)}$, where $\lambda_1(\Lambda)$ is the length of the shortest vector in the lattice $\Lambda$. Then, we show how this method can be used for solving SVP and the Closest Vector Problem (CVP) with approximation factor $\gamma=1.993$ in a $n$-dimensional lattice in time $3^{n+o(n)}$. Previous algorithms for enumerating take super-exponential running time with polynomial memory. For instance, Kannan algorithm takes time $n^{n/(2e)+o(n)}$, however ours also requires exponential memory and we propose different time/memory tradeoffs. Recently, Aggarwal, Dadush, Regev and Stephens-Davidowitz describe a randomized algorithm with running time $2^{n+o(n)}$ at STOC\u27 15 for solving SVP and approximation version of SVP and CVP at FOCS\u2715. However, it is not possible to use a time/memory tradeoff for their algorithms. Their main result presents an algorithm that samples an exponential number of random vectors in a Discrete Gaussian distribution with width below the smoothing parameter of the lattice. Our algorithm is related to the hill climbing of Liu, Lyubashevsky and Micciancio from RANDOM\u27 06 to solve the bounding decoding problem with preprocessing. It has been later improved by Dadush, Regev, Stephens-Davidowitz for solving the CVP with preprocessing problem at CCC\u2714. However the latter algorithm only looks for one lattice vector while we show that we can enumerate all lattice vectors in a ball. Finally, in these papers, they use a preprocessing to obtain a succinct representation of some lattice function. We show in a first step that we can obtain the same information using an exponential-time algorithm based on a collision search algorithm similar to the reduction of Micciancio and Peikert for the SIS problem with small modulus at CRYPTO\u27 13