Lattice Reduction Research Articles

Efficient detection in uniform linear and planar arrays MIMO systems under spatial correlated channels

SummaryIn this paper, the efficiency of various multiple‐input multiple‐output (MIMO) detectors was analyzed from the perspective of highly correlated channels, where MIMO systems have a lack of performance, besides in some cases, an increasing complexity. Considering this hard but a useful scenario, various MIMO detection schemes were accurately evaluated concerning complexity and bit error rate performance. Specifically, successive interference cancellation, lattice reduction, and the combination of them were associated with conventional linear MIMO detection techniques. To demonstrate effectiveness, a wide range of the number of antennas and modulation formats have been considered aiming to verify the potential of such MIMO detection techniques according to their performance‐complexity trade‐off. We have also studied the correlation effect when both transmit and receiver sides are equipped with uniform linear array and uniform planar array antenna configurations. The performance of different detectors is carefully compared when both antenna array configurations are deployed considering a different number of antennas and modulation order, especially under near‐massive MIMO condition. We have also discussed the relationship between the array factor and the bit error rate performance of both antenna array structures.

Volumes and distributions for random unimodular complex and quaternion lattices

Two themes associated with invariant measures on the matrix groups SLN(F), with F=R,C or H, and their corresponding lattices parametrised by SLN(F)/SLN(O), O being an appropriate Euclidean ring of integers, are considered. The first is the computation of the volume of the subset of SLN(F) with bounded 2-norm or Frobenius norm. Key here is the decomposition of measure in terms of the singular values. The form of the volume, for large values of the bound, is relevant to asymptotic counting problems in SLN(O). The second is the problem of lattice reduction in the case N=2. A unified proof of the validity of the appropriate analogue of the Lagrange–Gauss algorithm for computing the shortest basis is given. A decomposition of measure corresponding to the QR decomposition is used to specify the invariant measure in the coordinates of the shortest basis vectors. With F=C this allows for the exact computation of the PDF of the first minimum (for O=Z[i] and Z[(1+−3)/2]), and the PDF of the second minimum and that of the angle between the minimal basis vectors (for O=Z[i]). It also encodes the specification of fundamental domains of the corresponding quotient spaces. Integration over the latter gives rise to certain number theoretic constants, which are also present in the asymptotic forms of the PDFs of the lengths of the shortest basis vectors. Siegel's mean value gives an alternative method to compute the arithmetic constants, allowing in particular the computation of the leading form of the PDF of the first minimum for F=H and O the Hurwitz integers, for which direct integration was not possible.

Algorithm and Architecture of a Low-Complexity and High-Parallelism Preprocessing-Based K -Best Detector for Large-Scale MIMO Systems

As a branch of sphere decoding, the K -best method has played an important role in detection in large-scale multiple-input-multiple-output (MIMO) systems. However, as the numbers of users and antennas grow, the preprocessing complexity increases significantly, which is one of the major issues with the K -best method. To address this problem, this paper proposes a preprocessing algorithm combining Cholesky sorted QR decomposition and partial iterative lattice reduction (CHOSLAR) for K -best detection in a 64-quadrature amplitude modulation (QAM) 16 $\times$ 16 MIMO system. First, Cholesky decomposition is conducted to perform sorted QR decomposition. Compared with conventional sorted QR decomposition, this method reduces the number of multiplications by 25.1% and increases parallelism. Then, a constant-throughput partial iterative lattice reduction method is adopted to achieve near-optimal detection accuracy. This method further increases parallelism, reduces the number of matrix swaps by 45.5%, and reduces the number of multiplications by 67.3%. Finally, a sorting-reduced K -best strategy is used for vector estimation, thereby, reducing the number of comparators by 84.7%. This method suffers an accuracy loss of only approximately 1.44 dB compared with maximum likelihood detection. Based on CHOSLAR, this paper proposes a fully pipelined very-large-scale-integration architecture. A series of different systolic arrays and parallel processing units achieves an optimal tradeoff among throughput, area consumption, and power consumption. This architectural layout is obtained via TSMC 65-nm 1P9M CMOS technology, and throughput metrics of 1.40 Gbps/W (throughput/power) and 0.62 Mbps/kG (throughput/area) are achieved, demonstrating that the proposed system is much more efficient than state-of-the-art designs.

Investigation of Crystalline and Amorphous Forms of Aluminum and Its Alloys: Computational Modeling and Experiment

The purpose of this study is to investigate polycrystalline lattices of aluminum (Al) under the stress–strain conditions in all-atom molecular dynamics simulations and Al alloys using X-ray diffraction. Isothermal uniaxial tension and compression of these polycrystalline lattices showed no dislocation nucleation peaks, which correspond only to the Al monocrystal form. The best tensile and compressive resistance characteristics were observed for a material with the highest grain number ([Formula: see text]) due to the significant reduction of the face-centered cubic lattice in the metal structure. This process is mainly driven by the gradual elevation of the system’s kinetic energy. In the experiment, the amorphous Al alloys with higher manganese composition (20.5%) were investigated, matching the simulated amorphous structures. Overall, the results suggest that the increase in number of grains in Al lattices diminishes the stress–strain impact due to a more disordered atomic-scale (amorphous) metal composition.

A Lattice Basis Reduction Approach for the Design of Finite Wordlength FIR Filters

Many applications of finite impulse response (FIR) digital filters impose strict format constraints on the filter coefficients. Such requirements increase the complexity of determining optimal designs for the problem at hand. We introduce a fast and efficient method, based on the computation of good nodes for polynomial interpolation and Euclidean lattice basis reduction. Experiments show that it returns quasi-optimal finite wordlength FIR filters; compared to previous approaches it also scales remarkably well (length 125 filters are treated in $ 9 s). It also proves useful for accelerating the determination of optimal finite wordlength FIR filters.

A novel low complexity lattice reduction algorithm for MIMO detection

A new low complexity lattice reduction algorithm was proposed, namely, the sorted integer Gauss transformation (SIGT). The SIGT algorithm can be interpreted as minimizing the longest basis vector first and assure that there was no integer projection between any two basis vectors. By applying simulation over Rayleigh fading channels, it was demonstrated that the proposed SIGT algorithm can have almost the same bit error rate (BER) performance as the LLL algorithm, while the SIGT algorithm requires only about half iteration as the LLL algorithm and the running time of each iteration for both algorithms were similar to each other. It is concluded that the SIGT algorithm can achieve almost the bit error rate (BER) performance, while the SIGT requires fewer iterations than the LLL.

Study on the inversion of doped concentration induced by millisecond pulsed laser irradiation silicon-based avalanche photodiode.

In this paper, an experimental study of silicon-based avalanche photodiode (Si-APD) with millisecond pulse laser irradiation was carried out, and the C-V curve of Si-APD was obtained by using a semiconductor analyzer. Based on the single-side abrupt junction character of n+p, combined with the corresponding theoretical derivation, the doping concentration varying with the axial depth of damaged Si-APD was obtained by inverse computation. The lattice dislocation and junction reduction were the fundamental causes of the reduced doping concentration. The research results provide a new method for the study of the internal doping concentration for detectors with millisecond pulse laser damage.

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.

Volumes for $${\mathrm{SL}}_N({\mathbb {R}})$$ SL N ( R ) , the Selberg Integral and Random Lattices

There is a natural left and right invariant Haar measure associated with the matrix groups GL $${}_N(\mathbb {R})$$ and SL $${}_N(\mathbb {R})$$ due to Siegel. For the associated volume to be finite it is necessary to truncate the groups by imposing a bound on the norm, or in the case of SL $${}_N(\mathbb {R})$$ , by restricting to a fundamental domain. We compute the asymptotic volumes associated with the Haar measure for GL $${}_N(\mathbb {R})$$ and SL $${}_N(\mathbb {R})$$ matrices in the case that the singular values lie between $$R_1$$ and $$1/R_2$$ in the former, and that the 2-norm, or alternatively the Frobenius norm, is bounded by R in the latter. By a result of Duke, Rudnick and Sarnak, such asymptotic formulas in the case of SL $${}_N(\mathbb {R})$$ imply an asymptotic counting formula for matrices in SL $${}_N(\mathbb {Z})$$ . We discuss too the sampling of SL $${}_N(\mathbb {R})$$ matrices from the truncated sets. By then using lattice reduction to a fundamental domain, we obtain histograms approximating the probability density functions of the lengths and pairwise angles of shortest length bases vectors in the case $$N=2$$ and 3, or equivalently of shortest linearly independent vectors in the corresponding random lattice. In the case $$N=2$$ these distributions are evaluated explicitly.

Modular Algebraic Precoding Relay System Model Based on Lattice Reduction

According to the existing multi-antenna relay forwarding protocol, the relay matrix can be split into two parts—relay nodes forward and backward filters direct concatenation, then in actual channel matrix design, relay’s forward and backward channel conditions are not ideal, which make system bit error rate (BER) performance is poor. Based on this problem, this paper proposes a modular algebraic precoding relay system model based on lattice reduction. At first, this method re-builds the relay’s forward and backward channel matrices, which can make the channel satisfy the orthogonality characteristics. Then, a modified equivalent relay channel matrix is precoding, which solves tradition algorithms’ poor performance problem when channel orthogonality is not good, and based on this idea, two improved algorithms are proposed. These two new algorithms can achieve better BER performance under poor relay channel conditions, make each user’s performance average, and satisfy system fairness requirements. Simulation results show that comparing with tradition algorithms, this paper proposes a relay channel model which is based on the lattice reduction precoding method, it gains a certain bit error performance. Finally, the improved algorithms are compared and analyzed with the sorting-based algorithm. The results show that the proposed algorithms have better performance. Therefore, in order to ensure users fairness, the optimization algorithm can be used instead of the high complexity sorting-based algorithm in the actual system.

Transceiver Design for MIMO VLC Systems With Integer-Forcing Receivers

In this paper, we investigate the transceiver design for multiple-input multiple-output visible light communication (VLC) systems that employ the integer-forcing (IF) lattice decoding technique. To facilitate the joint design of the transmitter precoder, the integer matrix and the receiver equalizer, we first give a necessary and sufficient condition for the integer matrix to be invertible over 1-D lattice. Based on this condition, we then propose a new method for choosing the integer matrix which achieves better performance than the existing lattice reduction method. Moreover, taking into account two typical constraints in VLC, we optimize transmit and receive matrices using the conditional gradient method or the projected gradient method. Finally, the integer matrix and the transmit and receive matrices are jointly optimized in an iterative manner. Simulation results are provided to demonstrate the performance improvement achieved by the proposed framework.

Multiferroic properties of Sr-doped BiFeO3 nanoparticles

The effects of ‘Sr-doping’ and ‘particle size’ on multiferroic properties of BiFeO3 nanoparticles were studied. Structural analysis revealed no influence on the rhombohedral structure of BiFeO3 with Sr-doping in the synthesized nanoparticles. The particle size of Sr-doped BiFeO3 nanoparticles was tailored by varying calcination temperature. The crystallite size got reduced with Sr-doping and also on decreasing the calcination temperature. Lattice strain was observed to enhance with reducing particle size of Sr-doped BiFeO3 nanoparticles. Magnetic measurements confirmed enhancement in magnetization via suppression of spin cycloidal structure with Sr-doping owing to creation of oxygen vacancies, enhanced lattice strain and reduction in particle size. Dielectric analysis revealed deterioration of dielectric constant of BiFeO3 with Sr-doping due to the formation of oxygen vacancies. Further, on reducing particle size of Sr-doped BiFeO3 nanoparticles, both saturation magnetization and dielectric constant values registered increase. Lowering of the magnetic phase transition temperature with particle size of Sr-doped BiFeO3 nanoparticles indicated weakening of magnetic exchange interactions.

Solving the large-scale knapsack feasibility problem using a distributed computation approach to integer programming

The knapsack feasibility problems have been intensively studied both because of their immediate applications in industry and financial management, but more pronounced for theoretical reasons, as knapsack problems frequently occur by relaxation of various integer programming problems. In this work, the large-scale knapsack feasibility problem is divided into two subproblems. The first subproblem is transforming of the knapsack feasibility problem into a polytope judgement problem which is based on lattice basis reduction. In the next subproblem, a distributed implementation of Dang and Ye’s fixed-point iterative algorithm is introduced to solve the polytope judgement problem generated in the former subproblem. Compared with the branch and bound method, numerical results show that this distributed fixed-point method is effective.

Raman study of In Ga1−N (x = 0.32–0.9) films irradiated with Xe ions at room temperature and 773 K

Abstract Wurtzite InxGa1−xN (x = 0.32, 0.47, 0.7, 0.8, and 0.9) films grown on the GaN epilayers were irradiated with 5 MeV Xe ions to fluences of 3 × 1013 and 6 × 1013 cm−2 at room temperature (RT) and 773 K. Raman spectroscopy was used to study the effects on structure and electron carrier concentration in the irradiated materials. The results show that the irradiation induces lattice relaxation and reduction of the electron carrier concentration in the films, the extent of which increases and decreases, respectively, with increasing In content x in InxGa1−xN. Compared to RT irradiation, significant simultaneous defect recovery was observed during irradiation at 773 K up to a fluence of 3 × 1013 cm−2. Further irradiation to 6 × 1013 cm−2 leads to delamination of the In-rich InxGa1−xN films (x = 0.7, 0.8 and 0.9) from the GaN epilayers.

Dual QR decomposition in K-best sphere detection for Internet of Things networks

Internet of Things (IoT) in 5G communications is becoming more popular due to its robust features. Multiple input multiple output (MIMO) in IoT is playing a vital role nowadays with large and massive concepts. At the same time detection in such scenario is becoming complex day by day. Sphere decoding algorithms is one of the MIMO detection schemes used for achieving near-optimal maximum likelihood detection (MLD) performance. Preprocessing algorithms such as lattice reduction and QR-decomposition (QRD) has been widely used along with the variants of MIMO detection for achieving better bit error rate (BER) performance. This research paper proposes a novel dual QRD (DQRD) algorithm to obtain a universal low complex-lattice reduction aided (UL-LRA)—K-best sphere detection (KSD) for MIMO networks. The proposed algorithm is evaluated under additive white Gaussian noise flat fading and indoor task group ac (TGac-channel D) channel model. The results proved that the BER performance was enhanced as well as the average runtime complexity of LRA–KSD was reduced by 45 and 39% in a flat faded and channel D respectively compared to the optimal MLD using 16-QAM.

Boosted KZ and LLL Algorithms

There exist two issues among popular lattice reduction (LR) algorithms that should cause our concern. The first one is Korkine-Zolotarev (KZ) and Lenstra-Lenstra-Lovasz (LLL) algorithms may increase the lengths of basis vectors. The other is KZ reduction suffers much worse performance than Minkowski reduction in terms of providing short basis vectors, despite its superior theoretical upper bounds. To address these limitations, we improve the size reduction steps in KZ and LLL to set up two new efficient algorithms, referred to as boosted KZ and LLL, for solving the shortest basis problem (SBP) with exponential and polynomial complexity, respectively. Both of them offer better actual performance than their classic counterparts, and the performance bounds for KZ are also improved. We apply them to designing integer-forcing (IF) linear receivers for multi-input multi-output (MIMO) communications. Our simulations confirm their rate and complexity advantages.

Soot Oxidation Activity of Redox and Non-Redox Metal Oxides Synthesised by EDTA–Citrate Method

In the present study, redox (CeO2, SnO2, Pr6O11 and Mn3O4) and non-redox (Gd2O3, La2O3 ZrO2 and HfO2) metal oxides were successfully synthesised using the EDTA–citrate complexing method and tested for soot oxidation activity. The characterization of the metal oxides is carried out using FTIR, XRD, BET surface area, pore volume analyser, SEM and TEM. The redox nature and metal–oxygen bond information of the metal oxides are obtained from XPS analysis. In redox metal oxides, three critical parameters [lattice oxygen binding energy, reduction temperature and Δr (ionic size difference of the corresponding metal oxide oxidation states)] govern the soot oxidation activity. Among the redox metal oxide samples, Mn3O4 and Pr6O11 samples showed lower binding energy for oxygen (Oβ—529.4, 528.9 eV respectively), lower reduction temperature (Tα—317 and 512 °C respectively) and have smaller Δr value (9 pm and 17 pm respectively). Thus, displayed a better soot oxidation activity (T50 = 484 and 482 °C respectively) than compared to other redox metal oxides. Among the non-redox metal oxides, HfO2 sample displayed higher BET surface area (21.06 m2/g), lattice strain (0.0157), smaller ionic radius (58.2 pm) and higher relative surface oxygen ratio (58%) and thus resulted in a significantly better soot oxidation activity (T50 = 483 °C) than compared to other non-redox metal oxides.

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.

Attacking OpenSSL ECDSA with a small amount of side-channel information

In this work, we mount a lattice attack on the ECDSA signatures implemented by the latest version of OpenSSL which uses the windowed non-adjacent form method to implement the scalar multiplication. We first develop a new way of extracting information from the side-channel results of the ECDSA signatures. Just given a small fraction of the information about a side-channel result denoted as double-and-add chain, we take advantage of the length of the chain together with positions of two non-zero digits to recover information about the ephemeral key. Combining the information of both the most significant digits and the least significant bits, we are able to gain more information about the ephemeral key. The problem of recovering ECDSA secret key is then translated to the hidden number problem which can be solved by lattice reduction algorithms. Our attack is mounted to the series secp256k1 curve, and the result shows that 85 signatures would be enough to recover the secret key, which is better than the result that previous attack gained only utilizing the information extracted from the least significant bits, using about 200 signatures to recover the secret key.

On the optimality of lattices for the coppersmith technique

We investigate a method for finding small integer solutions of a univariate modular equation, that was introduced by Coppersmith (Proceedings of Eurocrypt 1996, LNCS, vol 1070, pp 155–165, 1996) and extended by May (New RSA vulnerabilities using lattice reduction methods, Ph.D. thesis, University of Paderborn, 2003). We will refer this method as the Coppersmith technique. This paper provides a way to analyze a general limitations of the lattice construction for the Coppersmith technique. Our analysis upper bounds the possible range of U that is asymptotically equal to the bound given by the original result of Coppersmith and May. This means that they have already given the best lattice construction. In addition, we investigate the optimality for the bivariate equation to solve the small inverse problem, which was inspired by Kunihiro’s (LNCS 7483:55–69, 2012) argument. In particular, we show the optimality for the Boneh–Durfee’s equation (Proceedings of Eurocrypt 1999, LNCS, vol 1592, pp 389–401, 1999) used for RSA cryptoanalysis, To show our results, we establish framework for the technique by following the relation of Howgrave-Graham (Proceedings of cryptography and coding, LNCS, vol 1355, pp 131–142, 1997), and then concretely define the conditions in which the technique succeed and fails. We then provide a way to analyze the range of U that satisfies these conditions. Technically, we show that the original result of Coppersmith achieves the optimal bound for U when constructing a lattice in the standard way. We then provide evidence which indicates that constructing a non-standard lattice is generally difficult.