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

*Paul Kirchner and Pierre-Alain Fouque*

**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' 15 for solving SVP and approximation version of SVP and CVP at FOCS'15. 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' 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'14. 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' 13.

**Category / Keywords: **public-key cryptography / cryptanalysis time-memory trade-off lattice shortest vector enumeration ball closest vector Kannan

**Date: **received 29 Feb 2016, last revised 29 Feb 2016

**Contact author: **paul kirchner at ens fr

**Available format(s): **PDF | BibTeX Citation

**Note: **Submitted to ICALP 2016.

**Version: **20160229:214205 (All versions of this report)

**Short URL: **ia.cr/2016/222

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