LLL Lattice Reduction Research Articles

Elevation Extraction and Deformation Monitoring by Multitemporal InSAR of Lupu Bridge in Shanghai

Monitoring, assessing, and understanding the structural health of large infrastructures, such as buildings, bridges, dams, tunnels, and highways, is important for urban development and management, as the gradual deterioration of such structures may result in catastrophic structural failure leading to high personal and economic losses. With a higher spatial resolution and a shorter revisit period, interferometric synthetic aperture radar (InSAR) plays an increasing role in the deformation monitoring and height extraction of structures. As a focal point of the InSAR data processing chain, phase unwrapping has a direct impact on the accuracy of the results. In complex urban areas, large elevation differences between the top and bottom parts of a large structure combined with a long interferometric baseline can result in a serious phase-wrapping problem. Here, with no accurate digital surface model (DSM) available, we handle the large phase gradients of arcs in multitemporal InSAR processing using a long–short baseline iteration method. Specifically, groups of interferometric pairs with short baselines are processed to obtain the rough initial elevation estimations of the persistent scatterers (PSs). The baseline threshold is then loosened in subsequent iterations to improve the accuracy of the elevation estimates step by step. The LLL lattice reduction algorithm (by Lenstra, Lenstra, and Lovász) is applied in the InSAR phase unwrapping process to rapidly reduce the search radius, compress the search space, and improve the success rate in resolving the phase ambiguities. Once the elevations of the selected PSs are determined, they are used in the following two-dimensional phase regression involving both elevations and deformations. A case study of Lupu Bridge in Shanghai is carried out for the algorithm’s verification. The estimated PS elevations agree well (within 1 m) with the official Lupu Bridge model data, while the PS deformation time series confirms that the bridge exhibits some symmetric progressive deformation, at 4–7 mm per year on both arches and 4–9 mm per year on the bridge deck during the SAR image acquisition period.

Success Probability of the Babai Estimators for Box-Constrained Integer Linear Models

In many applications including communications, one may encounter a linear model where the parameter vector $\hbx$ is an integer vector in a box. To estimate $\hbx$, a typical method is to solve a box-constrained integer least squares (BILS) problem. However, due to its high complexity, the box-constrained Babai integer point $\x^\sBB$ is commonly used as a suboptimal solution. In this paper, we first derive formulas for the success probability $P^\sBB$ of $\x^\sBB$ and the success probability $P^\sOB$ of the ordinary Babai integer point $\x^\sOB$ when $\hbx$ is uniformly distributed over the constraint box. Some properties of $P^\sBB$ and $P^\sOB$ and the relationship between them are studied. Then, we investigate the effects of some column permutation strategies on $\P^\sBB$. In addition to V-BLAST and SQRD, we also consider the permutation strategy involved in the LLL lattice reduction, to be referred to as LLL-P. On the one hand, we show that when the noise is relatively small, LLL-P always increases $P^\sBB$ and argue why both V-BLAST and SQRD often increase $P^\sBB$; and on the other hand, we show that when the noise is relatively large, LLL-P always decreases $P^\sBB$ and argue why both V-BLAST and SQRD often decrease $P^\sBB$. We also derive a column permutation invariant bound on $P^\sBB$, which is an upper bound and a lower bound under these two opposite conditions, respectively. Numerical results demonstrate our findings. Finally, we consider a conjecture concerning $\x^\sOB$ proposed by Ma et al. We first construct an example to show that the conjecture does not hold in general, and then show that it does hold under some conditions.

GfcLLL: A Greedy Selection-Based Approach for Fixed-Complexity LLL Reduction

The LLL lattice reduction has been widely used to decrease the bit error rate (BER) of the Babai point, but its running time varies much from matrix to matrix. To address this problem, some fixed-complexity LLL reductions (FCLLL) have been proposed. In this letter, we propose two greedy selection-based FCLLL algorithms: GfcLLL(1) and GfcLLL(2). Simulations show that both of them give Babai points with lower BER in similar or much shorter CPU time than existing ones.

Perturbation Analysis of the QR factor R in the context of LLL lattice basis reduction

In 1982, Arjen Lenstra, Hendrik Lenstra Jr. and László Lovász introduced an efficiently computable notion of reduction of basis of a Euclidean lattice that is now commonly referred to as LLL-reduction. The precise definition involves the R-factor of the QR factorization of the basis matrix. In order to circumvent the use of rational/exact arithmetic with large bit-sizes, it is tempting to consider using floating-point arithmetic with small precision to compute the R-factor. In the present article, we investigate the accuracy of the factor R of the QR factorisation of an LLL-reduced basis. Our main contribution is the first fully rigorous perturbation analysis of the R-factor of LLL-reduced matrices under columnwise perturbations. Our results are very useful to devise LLL-type algorithms relying on floating-point approximations.

Practical aspects of preprocessing techniques for K-Best tree search MIMO detectors

It has been shown in several works that some preprocessing techniques can improve data detection performance when they are applied to the channel matrix of MIMO wireless systems. In particular, these techniques can be used previously to K-Best tree search algorithms, and they are known to achieve successful results. Throughout this work, the performance and complexity of two preprocessing techniques (VBLAST ZF-DFE ordering and LLL lattice-reduction) are evaluated and compared. The LLL algorithm and a recently proposed fixed-complexity version of it are tested. In addition, a low-complexity implementation of the VBLAST ZF-DFE method is proposed. Results show that the LLL preprocessing is less costly than the VBLAST ZF-DFE ordering in average. Also, the BER curves of the K-Best detector in a 4 × 4 MIMO system reveal that the LLL method can only present better detection performance than the VBLAST ZF-DFE ordering for high SNRs and low values of K.

Cryptanalysis of RSA for a special case with d >e

In this paper, we study the RSA public key cryptosystem in a special case with the private exponent d larger than the public exponent e. When N0.258 ⩽ e ⩽ N0.854, d > e and satisfies the given conditions, we can perform cryptanalytic attacks based on the LLL lattice basis reduction algorithm. The idea is an extension of Boneh and Durfee’s researches on low private key RSA, and provides a new solution to finding weak keys in RSA cryptosystems.

Novel Joint Sorting and Reduction Technique for Delay-Constrained LLL-Aided MIMO Detection

In this letter, we introduce a novel technique to speed up the famous LLL lattice reduction algorithm by use of sorting. Simulation results reveal that when applied to LLL-reduction-aided MIMO detectors, our proposed joint sorting and reduction technique can find an LLL-reduced basis with the average number of basis vector swappings reduced by about 47.5% for a 2 times 2 system and more for higher dimensional systems, without degrading the detection performance. Moreover, when the maximum number of vector swapping is limited, our proposed algorithm significantly outperform the conventional one at low-to-moderate bit error rates. This suggests that it can be applied advantageously to those MIMO systems with a strict delay constraint.

Heuristics on lattice basis reduction in practice

In this paper we provide a survey on LLL lattice basis reduction in practice. We introduce several new heuristics as to speed up known lattice basis reduction methods and improve the quality of the computed reduced lattice basis in practice. We analyze substantial experimental data and to our knowledge, we are the first to present general heuristics for determining which variant of the reduction algorithm, for varied parameter choices, yields the most efficient reduction strategy for reducing a particular problem instance.