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N-Dimensional LLL Reduction Algorithm with Pivoted Reflection

School of Electronic Engineering, Beijing University of Posts and Telecommunications, No. 10 Xitucheng Road, Beijing 100876, China
Author to whom correspondence should be addressed.
Sensors 2018, 18(1), 283;
Received: 18 December 2017 / Revised: 12 January 2018 / Accepted: 14 January 2018 / Published: 19 January 2018
(This article belongs to the Section Remote Sensors, Control, and Telemetry)
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. View Full-Text
Keywords: LLL reduction; pivoted reflection; integer least squares (ILS); global navigation satellite system (GNSS) LLL reduction; pivoted reflection; integer least squares (ILS); global navigation satellite system (GNSS)
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Deng, Z.; Zhu, D.; Yin, L. N-Dimensional LLL Reduction Algorithm with Pivoted Reflection. Sensors 2018, 18, 283.

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