# ANALYSIS OF THE UPPER BOUND ON THE COMPLEXITY OF LLL ALGORITHM

• PARK, YUNJU (DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE, KOREA SCIENCE ACADEMY OF KOREA ADVANCED INSTITUTE OF SCIENCE AND TECHNOLOGY) ;
• PARK, JAEHYUN (DEPARTMENT OF ELECTRONIC ENGINEERING, PUKYONG NATIONAL UNIVERSITY)
• Accepted : 2016.05.25
• Published : 2016.06.25

#### Abstract

We analyze the complexity of the LLL algorithm, invented by Lenstra, Lenstra, and $Lov{\acute{a}}sz$ as a a well-known lattice reduction (LR) algorithm which is previously known as having the complexity of $O(N^4{\log}B)$ multiplications (or, $O(N^5({\log}B)^2)$ bit operations) for a lattice basis matrix $H({\in}{\mathbb{R}}^{M{\times}N})$ where B is the maximum value among the squared norm of columns of H. This implies that the complexity of the lattice reduction algorithm depends only on the matrix size and the lattice basis norm. However, the matrix structures (i.e., the correlation among the columns) of a given lattice matrix, which is usually measured by its condition number or determinant, can affect the computational complexity of the LR algorithm. In this paper, to see how the matrix structures can affect the LLL algorithm's complexity, we derive a more tight upper bound on the complexity of LLL algorithm in terms of the condition number and determinant of a given lattice matrix. We also analyze the complexities of the LLL updating/downdating schemes using the proposed upper bound.

#### Acknowledgement

Supported by : Pukyong National University

#### References

1. A.K. Lenstra, J.H.W. Lenstra, and L. Lovasz, Factoring polynomials with rational coefficients, Math. Annal., 261(4) (1982), 515-534. https://doi.org/10.1007/BF01457454
2. P. Nguyen and J. Stern, Lattice reduction in cryptology: An update, Algorithm. Numb. Theor.: 4th Intern. Symp. ANTS-IV, Lecture Notes in Comput. Sci., 1838 (2000), Springer, 85-112.
3. C.P. Schnorr, A hierarchy of polynomial time lattice basis reduction algorithms, Theor. Comput. Sci., 53 (1987), 201-224. https://doi.org/10.1016/0304-3975(87)90064-8
4. K. Lee, J. Chun, and L. Hanzo, Optimal lattice-reduction aided successive interference cancellation for mimo systems, IEEE Trans. Wir. Comm., 6(7) (2007), 2438-2443. https://doi.org/10.1109/TWC.2007.06058
5. C. Windpassinger and R.F.H. Fischer, Low-complexity near-maximum-likelihood detection and precoding for MIMO systems using lattice reduction, Proc. IEEE Inform. Theor. Workshop (ITW), (2003), 345-348.
6. H.Yao and G.W. Wornell, Lattice-reduction-aided detectors for MIMO communication systems, Proc. IEEE Glob. Telec. Conf., Taipei, Taiwan, 1 (2002), 424-428.
7. D.Wubben, R.Bohnke, V.Kuhn, and K.D. Kammeyer, Near-maximum-likelihood detection of MIMO systems using MMSE-based lattice-reduction, Proc. IEEE Int. Conf. Comm., 2 (2004), 798-802.
8. J. Park and Y. Park, Efficient lattice reduction updating and downdating methods and analysis, J. KSIAM, 19(2) (2015), 171-188.
9. M.C. Cary, Lattice basis reduction: Algorithms and applications, Unpublished draft available at https://www.researchgate.net/publication/265107426 Lattice Basis Reduction Algorithms and Applications, (2002).
10. G.H. Golub and C.F.V. Loan, Matrix Computations, 3rd ed. Baltimore: Johns Hopkins Univ. Press, (1996).
11. W. Backes and S. Wetzel, An efficient LLL gram using buffered transformations, Proc. Comput. Alg. Sci. Comp., LNCS2005, 4770 (2007), 31-44.
12. W. Backes and S. Wetzel, Heuristics on lattice basis reduction in practice, ACM J. Exp. Algor., 7 (2002).
13. D.G. Papachristoudis, S.T. Halkidis, and G. Stephanides, An experimental comparison of some LLL-type lattice basis reduction algorithms, Int. J. Appl. Comput. Math., 1(3) (2015), 327-342. https://doi.org/10.1007/s40819-014-0023-5