Lattice Reduction Algorithm Research Articles

A Low Complexity Diversity Achieving Decoder Based on a Two-Stage Lattice Reduction in Frequency-Selective MIMO Channels

Based on the V-BLAST zero-padded block transmission scheme in the frequency-selective multiple-input multiple-output channels, we propose a two-stage lattice-reduction (LR)-aided decoder and prove analytically that it can achieve the maximum diversity gain, which is equal to the product of the number of receive-antennas and the number of multipath channel taps. In the proposed decoding method, at first, we decouple the detection problem into several reduced-size detection problems and perform the lattice reduction algorithm for the decoupled blocks. Then, by choosing a proper threshold, a clustering algorithm divides the blocks into the maximum two clusters. We prove that the corresponding blocks of the first cluster achieve the full diversity through their linear equalization. For the remainder of the blocks, based on the entire observations of the second cluster, we propose an LR-aided detection method, which decodes the information symbols in this cluster with full diversity. Our proposed two-stage LR-aided algorithm performs better than the conventional LR-aided detection in terms of both the decoding computational complexity and the error probability. Additionally, we prove analytically that at high signal-to-noise ratio, the decoding complexity of the proposed decoder is equal to that of decoding the first cluster.

Analysing recursive preprocessing of BKZ lattice reduction algorithm

Lattice problems are considered as the key elements in many areas of computer science as well as in cryptography; the most important of which is the shortest vector problem and its approximate variants. Algorithms for this problem are known as lattice reduction algorithms. Currently, the most practical lattice reduction algorithm for such problems is the block Korkine–Zolotarev (BKZ) algorithm and its variants. The authors optimise both the pruning and the preprocessing parameters of the recursive (aborted, extreme pruned) preprocessing of the BKZ lattice reduction algorithm and improve the results from Asiacrypt'11 by Chen and Nguyen. The authors derive approximate closed-form complexity formulas (based on the sandpile model assumption model by Hanrot et al.) for the enumeration time which allow a simple estimation of complexity without running the simulation algorithm (by Chen and Nguyen) and asymptotically suggests a modified extreme pruning bounding profiles with different parameters. Hence, the authors’ contributions are in optimising and improving the analysis of the complexity upper bound estimates presented by Chen and Nguyen, based on the same recursive-BKZ preprocessing model.

Low-complexity lattice reduction algorithm for MIMO detectors with tree searching

In this paper, we propose a low-complexity lattice reduction (LR) algorithm for multiple-input multiple-output (MIMO) detectors with tree searching. Whereas conventional approaches are based exclusively on channel characteristics, we focus on joint optimisation by employing an early termination criterion in the context of MIMO detection. In this regard, incremental LR (ILR) was previously proposed. However, the ILR is limited to LR-aided successive interference cancellation (SIC) detectors which have considerable bit-error-rate (BER) performance degradation compared to optimal detectors. Hence, in this paper, we extend the conventional ILR to be applicable to the LR-aided detectors with near-optimal performance. Furthermore, we perform the hypothetical analysis and several novel modifications to handle the obstacles for the application of the ILR to LR-aided detectors other than the LR-aided SIC detectors. The simulation results demonstrate that the computational complexity is considerably reduced, with BER performance degradation of 10−5.

Low Complexity Coefficient Selection Algorithms for Compute-and-Forward

Compute-and-forward (C&F) has been proposed as an efficient strategy to reduce the backhaul load for distributed antenna systems. Finding the optimal coefficients in C&F has commonly been treated as a shortest vector problem, which is NP-hard. The point of our work and of Sahraei’s recent work is that the C&F coefficient problem can be much simpler. Due to the special structure of C&F, some low polynomial complexity optimal algorithms have recently been developed. However, these methods can be applied to real-valued channels and integer-based lattices only. In this paper, we consider the complex valued channel with complex integer-based lattices. For the first time, we propose a low polynomial complexity algorithm to find the optimal solution for the complex scenario. Then, we propose a simple linear search algorithm, which is conceptually suboptimal, and however, numerical results show that the performance degradation is negligible compared with the optimal method. Both algorithms are suitable for lattices over any algebraic integers, and significantly outperform the lattice reduction algorithm. The complexity of both algorithms is investigated both theoretically and numerically. The results show that our proposed algorithms achieve better performance-complexity tradeoffs compared with the existing algorithms.

A novel hybrid fix‐LLL lattice reduction algorithm for MIMO detection system

Lenstra–Lenstra–Lovász (LLL) is an effective lattice reduction algorithm for multiple‐input multiple‐output (MIMO) systems, which was considered to achieve full diversity in the MIMO fading channel. In the LLL algorithm, size reduction is performed for pairs of consecutive basis vectors, and the concern of numerical stability is raised. However, the whole complexity of the LLL algorithm is of polynomial order, and its characteristics of the convergence perform poor in MIMO system. In this paper, a variant version, named the novel hybrid algorithm, which combines both fix measurement and round measurement, is proposed. By modifying the iteration criterion and choosing proper values of the parameters, the algorithm has a large probability to skip the size reduction and cause a faster convergence, It means in one algorithm iteration the LLL potential can be reduced as much as possible. Also, the performance bound derived by the proximity factor shows that the hybrid LLL has a minor performance loss compared to the LLL algorithm. As a direct consequence, in simulation results, the hybrid LLL algorithm can make a better compromise between the rate of convergence, complexity of the algorithm, and algorithmic performance. © 2016 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc.

Efficient Integer Coefficient Search for Compute-and-Forward

Integer coefficient selection is an important decoding step in the implementation of compute-and-forward (C-F) relaying scheme. Choosing the optimal integer coefficients in C-F has been shown to be a shortest vector problem (SVP) which is known to be NP hard in its general form. Exhaustive search of the integer coefficients is only feasible in complexity for small number of users while approximation algorithms such as Lenstra-Lenstra-Lovasz (LLL) lattice reduction algorithm only find a vector within an exponential factor of the shortest vector. An optimal deterministic algorithm was proposed for C-F by Sahraei and Gastpar specifically for the real valued channel case. In this paper, we adapt their idea to the complex valued channel and propose an efficient search algorithm to find the optimal integer coefficient vectors over the ring of Gaussian integers and the ring of Eisenstein integers. A second algorithm is then proposed that generalises our search algorithm to the Integer-Forcing MIMO C-F receiver. Performance and efficiency of the proposed algorithms are evaluated through simulations and theoretical analysis.

Computationally efficient fixed complexity LLL algorithm for lattice‐reduction‐aided multiple‐input–multiple‐output precoding

In multiple-input–multiple-output broadcast channels, lattice reduction (LR) preprocessing technique can significantly improve the precoding performance. Among the existing LR algorithms, the fixed complexity Lenstra–Lenstra–Lovasz (fcLLL) algorithm applying limited number of LLL loops is suitable for the real-time communication system. However, fcLLL algorithm suffers from higher average complexity. Aiming at this problem, a computationally efficient fcLLL (CE-fcLLL) algorithm for LR-aided (LRA) precoding is developed in this study. First, the authors analyse the impact of fcLLL algorithm on the signal-to-noise ratio performance of LRA precoding by a power factor (PF) which is defined to measure the relation of reduced basis and transmit power of LRA precoding. Then, they propose a CE-fcLLL algorithm by designing a new LLL loop and introducing new early termination conditions to reduce redundant and inefficient LR operation in fcLLL algorithm. Finally, they define a PF loss factor to optimise the PF threshold and the number of LLL loops, which can lead to a performance-complexity tradeoff. Simulation results show that the proposed algorithm for LRA precoding can achieve better bit-error-rate performance than the fcLLL algorithm with remarkable complexity savings in the same upper complexity bound.

An effective QR-based reduction algorithm for the fast estimation of GNSS high-dimensional ambiguity resolution

To fast estimate high-dimensional ambiguities, we propose a new lattice reduction algorithm based on QR decomposition, which achieves fast integer transformation through an iterative strategy of whole size reduction and the deep insertion of minimum basis vectors. It acquires better basis vectors for ambiguity resolution. The feasibility of the proposed algorithm is verified by comparing its performance with those of LAMBDA, LLL, the parallel Cholesky-based reduction algorithm with ascending sorting (ASCE), and a modified LLL algorithm with deep insertions (PotLLL) under three experimental scenarios. The Hermite defect and defined length ratio are used to measure the reduction quality. Both metrics verify that our proposed method acquires a good reduced basis for accelerating the ambiguity search. To evaluate the practical ambiguity resolution behaviour, we tracked the runtime of ambiguity resolution. The results show that the computational efficiency of the proposed algorithm is better than those of the comparative algorithms.

Joint Lattice-Reduction-Aided Precoder Design for Multiuser MIMO Relay System

Lattice reduction (LR) has been used widely in conventional multiple-input multiple-output (MIMO) systems to enhance the performance. However, LR is hard to be applied to the relay systems which are important but more complicated in the wireless communication theory. This paper introduces a new viewpoint for utilizing LR in multiuser MIMO relay systems. The vector precoding (VP) is designed along with zero force (ZF) criterion and minimum mean square error (MMSE) criterion and enhanced by LR algorithm. This implementable precoder design combines nonlinear processing at the base station (BS) and linear processing at the relay. This precoder is capable of avoiding multiuser interference (MUI) at the mobile stations (MSs) and achieving excellent performance. Moreover, it is shown that the amount of feedback information is much less than that of the singular value decomposition (SVD) design. Simulation results show that the proposed scheme using the complex version of the Lenstra--Lenstra--Lovasz (LLL) algorithm significantly improves system performance.

An efficient lattice reduction using reuse technique blockwisely on NTRU

In this paper, we propose a lattice reduction algorithm for use with NTRU lattices. Given an NTRU lattice as its input, the algorithm computes an LLL-reduced basis. The proposed lattice reduction algorithm is more efficient than the classical LLL algorithm. Recently, a lattice reduction algorithm for ideal lattices, named iLLL, was proposed by Plantard, Susilo, and Zhang. This algorithm is identical to that of the LLL except for the fact that it contains an additional subroutine, named Reuse. The subroutine serves to further reduce a set of short vectors that has already been computed by the algorithm prior to its initiation. As a result, the iLLL is able to output an LLL-reduced basis more efficiently than the LLL is able to do so. However, the iLLL cannot be directly applied to an NTRU lattice, because it is not an ideal lattice. Yet, from the fact that an NTRU lattice is also a module lattice (a generalization of an ideal lattice), we can adapt the main idea behind the iLLL blockwisely in our approach to NTRU lattices. We demonstrate that the proposed algorithm (containing a modified version of the aforementioned subroutine Reuse) is asymptotically n2 times faster at outputting an LLL-reduced basis than the LLL when applied to NTRU lattices of dimension n. In the case of small n, our experiments show that the proposed algorithm is slightly faster at outputting an LLL-reduced basis than the LLL. In addition, we present an example of how to recover a private key of an NTRU encryption scheme by using the proposed algorithm in the case of n=22.

ANALYSIS OF THE UPPER BOUND ON THE COMPLEXITY OF LLL ALGORITHM

We analyze the complexity of the LLL algorithm, invented by Lenstra, Lenstra, and <TEX>$Lov{\acute{a}}sz$</TEX> as a a well-known lattice reduction (LR) algorithm which is previously known as having the complexity of <TEX>$O(N^4{\log}B)$</TEX> multiplications (or, <TEX>$O(N^5({\log}B)^2)$</TEX> bit operations) for a lattice basis matrix <TEX>$H({\in}{\mathbb{R}}^{M{\times}N})$</TEX> 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.

An improved scheme based on log-likelihood-ratio for lattice reduction-aided MIMO detection

Lattice reduction (LR)-aided detectors have been shown great potentials in wireless communications for their low complexity and low bit-error-rate (BER) performance. The LR algorithms use the unimodular transformation to improve the orthogonality of the channel matrix. However, the LR algorithms only utilize the channel state information (CSI) and do not take account for the received signal, which is also important information in enhancing the performance of the detectors. In this paper, we make a readjustment of the received signal in the LR domain and propose a new scheme which is based on the log-likelihood-ratio (LLR) criterion to improve the LR-aided detectors. The motivation of using the LLR criterion is that it utilizes both the received signal and the CSI, so that it can provide exact pairwise error probabilities (PEPs) of the symbols. Then, in the proposed scheme, we design the LLR-based transformation algorithm (TA) which uses the unimodular transformation to minimize the PEPs of the symbols by the LLR criterion. Note that the PEPs of the symbols affect the error propagation in the vertical Bell Laboratories Layered Space-Time (VBLAST) detector, and decreasing the PEPs can reduce the error propagation in the VBLAST detectors; thus, our LLR-based TA-aided VBLAST detectors will exhibit better BER performance than the previous LR-aided VBLAST detectors. Both the BER performance and the computational complexity are demonstrated through the simulation results.

On the concrete hardness of Learning with Errors

Abstract The learning with errors (LWE) problem has become a central building block of modern cryptographic constructions. This work collects and presents hardness results for concrete instances of LWE. In particular, we discuss algorithms proposed in the literature and give the expected resources required to run them. We consider both generic instances of LWE as well as small secret variants. Since for several methods of solving LWE we require a lattice reduction step, we also review lattice reduction algorithms and use a refined model for estimating their running times. We also give concrete estimates for various families of LWE instances, provide a Sage module for computing these estimates and highlight gaps in the knowledge about algorithms for solving the LWE problem.

Lattice reduction‐ordered successive interference cancellation detection algorithm for multiple‐input–multiple‐output system

Lattice reduction (LR) is a powerful technique for improving the performance of linear multiple-input–multiple-output detection methods. The efficient LR algorithms can largely improve the performance of the linear detectors (LDs). Note that the ordered successive interference cancellation (OSIC) system can decrease the interference between antennas and provide performance gain of the LDs. In this paper, a novel LR-aided algorithm called NLR-OSIC improving the performance of the OSIC system has been proposed. Most existing LR algorithms are designed to improve the orthogonality of channel matrices, which is not directly related to the error performance of the OSIC system. While the authors’ algorithm maximises the signal-to-interference-plus-noise ratio (SINR) of the detected symbol in each stage of the OSIC system, thus exhibiting improved error rate than the previous LR-aided LDs and their corresponding OSIC algorithms. In each stage, the authors verify that maximising the SINR of the detected symbol can be formulated as a shortest vector problem which is solved by a suboptimal algorithm in this study. In the end of this study, the error rate performance of the proposed algorithm as well as the required complexity has been demonstrated through extensive computer simulations.

An Experimental Comparison of Some LLL-Type Lattice Basis Reduction Algorithms

In this paper we experimentally compare the performance of the $$L^2$$ lattice basis reduction algorithm, whose importance recently became evident, with our own Gram-based lattice basis reduction algorithm, which is a variant of the Schnorr–Euchner algorithm. We conclude with observations about the algorithms under investigation for lattice basis dimensions up to the theoretical limit. We also reexamine the notion of “buffered transformations” and its impact on performance of lattice basis reduction algorithms. We experimentally compare four different algorithms directly in the Sage Mathematics Software: our own algorithm, the $$L^2$$ algorithm and “buffered” versions of them resulting in a total of four algorithms.

Cooperative transmission schemes for energy‐efficient collaborative wireless sensor networks

Energy conservation is one of the prime concerns that leads the researcher to investigate collaborative wireless sensor networks with some application specific challenges. Such challenges include combining distributed data synchronously, performing power aware signal processing, defining communication methods that can provide progressive accuracy and, optimising processing and communication for signal transmission. A cooperative resource selection and transmission scheme is proposed to improve the performance of collaborative wireless sensor networks in terms of maintaining link reliability. A measure of Channel Quality Index (CQI) is also proposed to obtain dynamic adaptivity and to optimise resource usage within wireless sensor networks according to environment conditions. As part of the proposed cooperative nature of transmission, the recently proposed transmit-receive antenna selection scheme and lattice reduction algorithm have also been considered. It is assumed that channel state information (CSI) is estimated at receiver and also there is a feedback link between the wireless sensing nodes and the fusion centre receiver. From the simulation results it is observed that for 99.99% detection reliability, the proposed adaptive transmission scheme and proposed hybrid scheme consume only 15% and 18% of energy respectively as compared to the conventional cooperative transmission.

A new parallel lattice reduction algorithm for BKZ reduced bases

In order to implement the original BKZ algorithm in parallel, we describe it in terms of parallelism and give its parallel implementation scheme. Then we analyze the efficiency of algorithm’s parallel implementation and show that the speedup factor of BKZ algorithm in parallel is extremely low. Therefore we present a new parallel lattice reduction algorithm suitable for multiprocessor computer architecture. The new algorithm can obtain a BKZ reduced basis and the parallel speedup is effective. Also with the practical results, although the computational complexity increases compared with the original BKZ algorithm, we still indicate that the new algorithm performs well in parallel and the time cost in parallel is less. At the same time, we show that the length of the shortest vector is smaller.

A Mathematical modeling and Simulation Analysis of Lattice reduction algorithms for large MIMO detections

Abstract – This paper analyses the performance of Multiple Input Multiple Output (MIMO) wireless communication system by combining it with an efficient algorithm Diagonal Reduction (DR). DR is a powerful tool to achieve more efficiently a high performance with less complexity. This paper proposes a DR algorithm in order to reduce its complexity named Greedy Diagonal Reduction (GDR) algorithm which gives reduced complexity with efficient performance at the receiver using the MATLAB communication tool box version 7.0 as a simulation tool. From the simulation results of various reduction algorithms it is observed that, DR can reduce the number of iterations using size reduction operations. Proposed DR algorithm gives identical Bit Error Rate (BER) performance with LLL algorithms when applied to Successive Interference cancellation (SIC) decoding. Greedy DR reduces the computational complexity in Multiple Input Multiple Output systems by improving the efficiency in terms of size reduction operations. Sphere Decoding (SD) technique gives better BER performance on comparing with Successive Interference cancellation decoding technique.

A Lattice Reduction-Aided MIMO Channel Equalizer in 90 nm CMOS Achieving 720 Mb/s

In this paper, a VLSI implementation of a complete MIMO channel equalization ASIC based on lattice reduction-aided linear detection is presented. The architecture performs preprocessing steps at channel rate and low-complexity linear data detection at symbol rate. Preprocessing is based on Seysen's algorithm for lattice reduction. We present algorithmic improvements of the lattice reduction preprocessing in terms of area and throughput of the VLSI implementation with minor impact on the error-rate. Due to the low-complexity implementation of the lattice reduction-aided data detection stage, our architecture is able to achieve very low power in typical packet-based MIMO wireless data transmission scenarios. The final 90 nm CMOS ASIC achieves an energy efficiency for the detection of 24 pJ/bit at a throughput of 720 Mbps with near-optimal error-rate performance.

Full-Diversity Approximated Lattice Reduction Algorithm for Low-Complexity MIMO Detection

In this letter, we propose a new approximated basis vector reordering (ABVR) criterion for low-complexity lattice reduction aided (LRA) multiple-input multiple-output (MIMO) detection. Despite the approximation, the ABVR criterion is proved to collect the full receiving diversity for LRA linear detection. A variant of the well-known complex Lenstra Lenstra Lovász (CLLL) algorithm, i.e., LLL with deep insertion (DLLL), is employed to accommodate the ABVR criterion (DLLL-ABVR). Compared with the original CLLL and other approximated algorithms, the proposed DLLL-ABVR algorithm largely reduces the number of basis vector reordering (BVR) operations. Simulation results show that, on a practical MIMO scale, the proposed lattice reduction algorithm provides similar detection performance, especially for successive interference cancelation (SIC) detectors, while requiring significantly lower computational complexity.