**Approximate Algorithms on Lattices with Small Determinant**

*Jung Hee Cheon and Changmin Lee*

**Abstract: **In this paper, we propose approximate lattice algorithms for
solving the shortest vector problem (SVP) and the closest vector
problem (CVP) on an $n$-dimensional Euclidean integral lattice
L. Our algorithms run in polynomial time of the dimension and
determinant of lattices and improve on the LLL algorithm when the
determinant of a lattice is less than 2^{n^2/4}. More precisely,
our approximate SVP algorithm gives a lattice vector of size \le
2^{\sqrt{\log\det L}} and our approximate CVP algorithm gives a
lattice vector, the distance of which to a target vector is
2^{\sqrt{\log\det L}} times the distance from the target vector
to the lattice. One interesting feature of our algorithms is that
their output sizes are independent of dimension and become smaller
as the determinant of L becomes smaller. For example, if \det
L=2^{n \sqrt n}, a short vector outputted from our approximate
SVP algorithm is of size 2^{n^{3/4}}, which is asymptotically
smaller than the size 2^{n/4+\sqrt n} of the outputted short
vectors of the LLL algorithm. It is similar to our approximate CVP
algorithm.

**Category / Keywords: **LLL algorithm, SVP, CVP, Hermite normal form, Babai's nearest plane algorithm

**Date: **received 13 May 2015, last revised 17 Jan 2016

**Contact author: **cocomi11 at snu ac kr

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

**Version: **20160118:053557 (All versions of this report)

**Short URL: **ia.cr/2015/461

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