N-Dimensional LLL Reduction Algorithm with Pivoted Reflection

The Lenstra-Lenstra-Lovász (LLL) lattice reduction algorithm and many of its variants have been widely used by cryptography, multiple-input-multiple-output (MIMO) communication systems and carrier phase positioning in global navigation satellite system (GNSS) to solve the integer least squares (ILS) problem. In this paper, we propose an n-dimensional LLL reduction algorithm (n-LLL), expanding the Lovász condition in LLL algorithm to n-dimensional space in order to obtain a further reduced basis. We also introduce pivoted Householder reflection into the algorithm to optimize the reduction time. For an m-order positive definite matrix, analysis shows that the n-LLL reduction algorithm will converge within finite steps and always produce better results than the original LLL reduction algorithm with n > 2. The simulations clearly prove that n-LLL is better than the original LLL in reducing the condition number of an ill-conditioned input matrix with 39% improvement on average for typical cases, which can significantly reduce the searching space for solving ILS problem. The simulation results also show that the pivoted reflection has significantly declined the number of swaps in the algorithm by 57%, making n-LLL a more practical reduction algorithm.

[1]  Peiliang Xu.  Random simulation and GPS decorrelation , 2001 .

[2]  László Lovász,et al.  Factoring polynomials with rational coefficients , 1982 .

[3]  Ron Hatch,et al.  Instantaneous Ambiguity Resolution , 1991 .

[4]  Damien Stehlé,et al.  Floating-Point LLL Revisited , 2005, EUROCRYPT.

[5]  Markus Gerke,et al.  Accuracy analysis of photogrammetric UAV image blocks: influence of onboard RTK-GNSS and cross flight patterns , 2016 .

[6]  M. Martin-Neira,et al.  Carrier Phase Ambiguity Resolution in GNSS-2 , 1997 .

[7]  Stephen P. Boyd,et al.  Integer parameter estimation in linear models with applications to GPS , 1996, Proceedings of 35th IEEE Conference on Decision and Control.

[8]  C. P. Schnorr,et al.  A Hierarchy of Polynomial Time Lattice Basis Reduction Algorithms , 1987, Theor. Comput. Sci..

[9]  K.-D. Kammeyer,et al.  MMSE extension of V-BLAST based on sorted QR decomposition , 2003, 2003 IEEE 58th Vehicular Technology Conference. VTC 2003-Fall (IEEE Cat. No.03CH37484).

[10]  Amir K. Khandani,et al.  LLL Reduction Achieves the Receive Diversity in MIMO Decoding , 2006, IEEE Transactions on Information Theory.

[11]  J. P. Provenzano,et al.  Implementation of a European infrastructure for satellite navigation: from system design to advanced technology evaluation , 2000, Int. J. Satell. Commun. Netw..

[12]  P. Teunissen.  The least-squares ambiguity decorrelation adjustment: a method for fast GPS integer ambiguity estimation , 1995 .

[13]  Peter Teunissen,et al.  On the Spectrum of the GPS DD-Ambiguities , 1994 .

[14]  E. Grafarend.  Mixed Integer-Real Valued Adjustment (IRA) Problems: GPS Initial Cycle Ambiguity Resolution by Means of the LLL Algorithm , 2000, GPS Solutions.

[15]  Felix Fontein,et al.  PotLLL: a polynomial time version of LLL with deep insertions , 2012, Designs, Codes and Cryptography.

[16]  P. Teunissen,et al.  The least-squares ambiguity decorrelation adjustment: its performance on short GPS baselines and short observation spans , 1997 .

[17]  X. Chang,et al.  MLAMBDA: a modified LAMBDA method for integer least-squares estimation , 2005 .

[18]  Wai Ho Mow,et al.  Complex Lattice Reduction Algorithm for Low-Complexity MIMO Detection , 2006, ArXiv.

[19]  Jung-Ha Kim,et al.  Research into navigation Algorithm for unmanned ground vehicle using Real Time Kinemtatic (RTK)-GPS , 2009, 2009 ICCAS-SICE.

[20]  László Babai,et al.  On Lovász’ lattice reduction and the nearest lattice point problem , 1986, Comb..

[21]  P. Xu.  Voronoi cells, probabilistic bounds, and hypothesis testing in mixed integer linear models , 2006, IEEE Transactions on Information Theory.

[22]  Peiliang Xu.  Parallel Cholesky-based reduction for the weighted integer least squares problem , 2011, Journal of Geodesy.

[23]  Khairul Nizam Tahar,et al.  Unmanned Aerial Vehicle Photogrammetric Results Using Different Real Time Kinematic Global Positioning System Approaches , 2013 .

[24]  Peter A. Hoeher,et al.  Fixed Complexity LLL Algorithm , 2009, IEEE Transactions on Signal Processing.

[25]  Claus-Peter Schnorr,et al.  Lattice basis reduction: Improved practical algorithms and solving subset sum problems , 1991, FCT.

[26]  Chris Rizos,et al.  The International GNSS Service in a changing landscape of Global Navigation Satellite Systems , 2009 .

[27]  X. Chang,et al.  On ‘decorrelation’ in solving integer least-squares problems for ambiguity determination , 2014 .

[28]  Xiao-Wen Chang,et al.  GfcLLL: A Greedy Selection-Based Approach for Fixed-Complexity LLL Reduction , 2016, IEEE Communications Letters.