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This paper proposes a novel approach to low complexity soft input decoding for lattice reduction-aided MIMO receivers. The proposed approach feeds a soft input decoder with soft signals made from hard decision signals generated by using a lattice reduction-aided linear detector. The soft signal is a weighted-sum of some candidate vectors that are near by the hard decision signal coming out from the lattice reduction-aided linear detector. This paper proposes a technique to adjust the weight adapt to the channel for the higher transmission performance. Furthermore, we propose to introduce a coefficient that is used for the weights in order to enhance the transmission performance. The transmission performance is evaluated in a 4×4 MIMO channel. When a linear MMSE filter or a serial interference canceller is used as the linear detector, the proposed technique achieves about 1.0dB better transmission performance at the BER of 10-5 than the decoder fed with the hard decision signals. In addition, the low computational complexity of the proposed technique is quantitatively evaluated.

Inspired by the advancements in (fully) homomorphic encryption in recent decades and its practical applications, we conducted a preliminary study on the underlying mathematical structure of the corresponding schemes. Hence, this paper focuses on investigating the challenge of deducing bivariate polynomials constructed using homomorphic operations, namely repetitive additions and multiplications. To begin with, we introduce an approach for solving the previously mentioned problem using Lagrange interpolation for the evaluation of univariate polynomials. This method is well-established for determining univariate polynomials that satisfy a specific set of points. Moreover, we propose a second approach based on modular knapsack resolution algorithms. These algorithms are designed to address optimization problems in which a set of objects with specific weights and values is involved. Finally, we provide recommendations on how to run our algorithms in order to obtain better results in terms of precision.

The crucial step in elliptic curve scalar multiplication based on scalar decompositions using efficient endomorphisms—such as GLV, GLS or GLV+GLS—is to produce a short basis of a lattice involving the eigenvalues of the endomorphisms, which usually is obtained by lattice basis reduction algorithms or even more specialized algorithms. Recently, lattice basis reduction is found to be unnecessary. Benjamin Smith (AMS 2015) was able to immediately write down a short basis of the lattice for the GLV, GLS, GLV+GLS of quadratic twists using elementary facts about quadratic rings. Certainly it is always more convenient to use a ready-made short basis than to compute a new one by some algorithm. In this paper, we extend Smith's method on GLV+GLS for quadratic twists to quartic and sextic twists, and give ready-made short bases for $ 4 $-dimensional decompositions on these high degree twisted curves. In particular, our method gives a unified short basis compared with Hu et al.'s method (DCC 2012) for $ 4 $-dimensional decompositions on sextic twisted curves.

International audience; The best lattice reduction algorithm known in theory for approximating the Shortest Vector Problem (SVP) over lattices is the slide reduction algorithm (STOC '08 & CRYPTO '20). In this paper, we first improve the running time analysis of computing slide-reduced bases based on potential functions. This analysis applies to a generic slide reduction algorithm that includes (natural variants of) slide reduction and block-Rankin reduction (ANTS '14). We then present a rigorous dynamic analysis of generic slide reduction using techniques originally applied to a variant of BKZ (CRYPTO '11). This provides guarantees on the quality of the current lattice basis during execution. This dynamic analysis not only implies sharper convergence for these algorithms to find a short nonzero vector (rather than a fully reduced basis), but also allows to heuristically model/trace the practical behaviour of slide reduction. Interestingly, this dynamic analysis inspires us to introduce a new slide reduction variant with better time/quality trade-offs. This is confirmed by both our experiments and simulation, which also show that our variant is competitive with state-of-the-art reduction algorithms. To the best of our knowledge, this work is the first attempt of improving the practical performance of slide reduction beyond speeding up the SVP oracle.

Lattice problems are algorithmic problems for lattices, whose hardness assures the security of lattice-based cryptography. Given a basis of a lattice, the goal of lattice basis reduction is to seek a new basis of the same lattice whose basis vectors are relatively short and nearly orthogonal to each other. Such reduction algorithms give a fundamental tool to solve lattice problems such as the shortest and the closest vector problems. In this paper, we provide a new self-dual variant of the DeepBKZ reduction algorithm. Our variant dynamically switches performing primal and dual reductions for local blocks of an input basis to effectively reduce the whole basis. In particular, we introduce a parameter of free dimensions to determine the sizes of local blocks to be reduced. We also implement our self-dual DeepBKZ to report its experimental results for random lattice bases.

The circular coordinates algorithm of de Silva, Morozov, and Vejdemo-Johansson takes as input a dataset together with a cohomology class representing a $1$-dimensional hole in the data; the output is a map from the data into the circle that captures this hole, and that is of minimum energy in a suitable sense. However, when applied to several cohomology classes, the output circle-valued maps can be "geometrically correlated" even if the chosen cohomology classes are linearly independent. It is shown in the original work that less correlated maps can be obtained with suitable integer linear combinations of the cohomology classes, with the linear combinations being chosen by inspection. In this paper, we identify a formal notion of geometric correlation between circle-valued maps which, in the Riemannian manifold case, corresponds to the Dirichlet form, a bilinear form derived from the Dirichlet energy. We describe a systematic procedure for constructing low energy torus-valued maps on data, starting from a set of linearly independent cohomology classes. We showcase our procedure with computational examples. Our main algorithm is based on the Lenstra--Lenstra--Lov\'asz algorithm from computational number theory., Comment: 24 pages, 12 figures. To appear in proceedings of 39th International Symposium on Computational Geometry

In this paper we study the linear congruential generator on elliptic curves from the cryptographic point of view. We show that if sufficiently many of the most significant bits of the composer and of three consecutive values of the sequence are given, then one can recover the seed and the composer (even in the case where the elliptic curve is private). The results are based on lattice reduction techniques and improve some recent approaches of the same security problem. We also estimate limits of some heuristic approaches, which still remain much weaker than those known for nonlinear congruential generators. Several examples are tested using implementations of ours algorithms.

The large scale multiuser multiple input multiple output (MU-MIMO) is one of the promising communication technology for 5G wireless networks as it offers reliability, high spectral efficiency and high throughput. The lattice reduction (LR) precoding based user level local likelihood ascent search (ULAS) detection scheme is proposed in this paper for efficient signal detection in large scale MU-MIMO system. The initial solution of ULAS algorithm is obtained from the LR precoding assisted zero forcing detector. The LR precoding transforms the non-orthogonal channel matrix into nearly orthogonal channel, which helps to mitigate inter antenna interference (IAI) exists at each user. The remaining multiuser interference (MUI) imposed to each user from undesired users is cancelled by the proposed ULAS multiuser detection scheme. Thus, the proposed LR precoding assisted ULAS mitigates both IAI and MUI unlike the classical detector, those try to moderate either IAI or MUI. By contrast, the proposed ULAS detector provides performance close to optimal maximum likelihood detector with just a fraction of its complexity.

We consider the problem of communicating over a relay-assisted multiple-input multiple-output (MIMO) channel with additive noise, in which physically separated relays forward quantized information to a central decoder where the transmitted message is to be decoded. We assume that channel state information is available in the transmitter and show that the design of a rational-forcing precoder—a precoder which is matched to the quantizers used in the relays—is beneficial for reducing the symbol error probability. It turns out that for such rational-forcing precoder based systems, there is natural tradeoff between the peak to average power ratio in the transmitter and the rate of communication between the relays and the central decoder. The precoder design problem is formulated mathematically, and several algorithms are developed for realizing this tradeoff. Optimality of the decoder communication rate is shown based on a result in distributed function computation. Numerical and simulation results show that a useful tradeoff can be obtained between the excess decoder communication rate and the peak-average power ratio in the transmitter.

Low-complexity lattice reduction algorithms are generally not optimized for solving the shortest basis problem (SBP). We fill this blank by tweaking the recently introduced sequential reduction (SR). In a quest for developing a provable and low-complexity reduction algorithm under the SR framework, we propose to employ successive interference cancellation (SIC) as a subroutine inside SR, and the whole algorithm is referred to as SR-SIC. On the theoretical front, we prove that the upper bound on the basis length of SR-SIC is better than those of major Lenstra, Lenstra, and Lovasz (LLL) based variants when the dimension of the lattice basis is no larger than 4. In practice, we show by simulations that SR-SIC yields higher information rate than LLL when designing integer-forcing linear receivers, in which SR-SIC also enjoys lower computational complexity.

We present a principled technique for reducing the lattice and matrix size in some applications of Coppersmith's lattice method for finding roots of modular polynomial equations. Motivated by ideas from machine learning, it relies on extrapolating patterns from the actual behavior of Coppersmith's attack for smaller parameter sizes, which can be thought of as "focus group" testing. When applied to the small-exponent RSA problem, our technique reduces lattice dimensions and consequently running times, and hence can be applied to a wider range of exponents. Moreover, in many difficult examples our attack is not only faster but also more successful in recovering the RSA secret key. We include a discussion of subtleties concerning whether or not existing metrics (such as enabling condition bounds) are decisive in predicting the true efficacy of attacks based on Coppersmith's method. Finally, indications are given which suggest certain lattice basis reduction algorithms (such as Nguyen-Stehl\'e's L2) may be particularly well-suited for Coppersmith's method., Comment: 21 pages, 5 figures

Computing HNF has a long history, but designing a storage efficient algorithm is a challenging issue for matrices of large sizes. One of the main challenges in the design of storage efficient HNF algorithm is to control the rank and the size of the intermediate input. In our proposed algorithm, we use a multiple extended gcd algorithm and show that the rank of the intermediate input matrix decreases as the number of iteration increases. The determinant of the intermediate input matrix of our algorithm is a factor of the determinant d of the input matrix and thus size reduction modulo d can be done in the computations of our algorithm. By using a lattice reduction algorithm and a proven quality of LLL reduced basis, we prove that the storage of the intermediate input matrix B k of our algorithm is less than ( n − k + 1 ) ( ( n − k + 1 ) ( n − k ) 4 + log 2 ⁡ d ) in bits. Therefore, it is expected that a smaller storage for kth iteration is required as k closes to n. We compare the intermediate computations of our algorithm to MW-type algorithm which has an optimal asymptotic space requirement. Our experimental example and results on intermediate size suggest that the intermediate storage of our HNF algorithm is comparable to MW-type algorithm and well controlled by the intermediate input size.

This article presents an $8\times 8$ lattice-reduction-aided (LRA) soft-output multiple-input multiple-output (MIMO) detector for Chinese enhanced ultrahigh throughput (EUHT) wireless local area network (LAN) standard. The preprocessing algorithm combining simplified-sorting Cholesky decomposition and low-complexity decoupled lattice reduction (LDLR) is proposed to reduce computational complexity and latency with parallelism improvement. In addition, K-best detection adopts a sorting-reduced strategy utilizing approximate ordered sequence. Compared with other published LRA K-best detection algorithms, simulation results show that our proposed algorithm has performance improvement. In addition, in order to save hardware resources, a folded K-best architecture and an optimized intermediate storage strategy are introduced. Furthermore, a fully pipelined VLSI architecture is designed in Semiconductor Manufacturing International Corporation (SMIC) 40-nm 1P9M technology to support the $8\times 8.64$ -QAM MIMO-OFDM system. The detector can achieve 3.85-Gb/s data throughput at 641-MHz clock frequency with 0.71- $\mu \text{s}$ latency. The proposed detector is competitive in terms of latency, throughput, and area efficiency to state-of-the-art works and can meet the data-rate requirement of the EUHT standard.

The lattice size of a lattice polytope P was defined and studied by Schicho, and Castryck and Cools. They provided an “onion skins” algorithm for computing the lattice size of a lattice polygon P in $$\mathbb R^2$$ based on passing successively to the convex hull of the interior lattice points of P. We explain the connection of the lattice size to the successive minima of $$K=(P+(-P))^*$$ and to the lattice reduction with respect to the general norm that corresponds to K. It follows that the generalized Gauss algorithm of Kaib and Schnorr (which is faster than the “onion skins” algorithm) computes the lattice size of any convex body in $$\mathbb R^2$$ . We extend the work of Kaib and Schnorr to dimension three, providing a fast algorithm for lattice reduction with respect to the general norm defined by a convex origin-symmetric body $$K\subset \mathbb R^3$$ . We also explain how to recover the successive minima of K and the lattice size of P from the obtained reduced basis and therefore provide a fast algorithm for computing the lattice size of any convex body $$P\subset \mathbb R^3$$ .

Massive multi-user multiple-input multiple-output (MU-MIMO) communication is a crucial technique for next-generation wireless systems because it enables high-throughput and ultra-reliable data transmission. However, high power consumption is a challenging problem in conventional transceiver designs, where each RF chain at the transmitter requires a pair of high-resolution digital-to-analog converters (DACs), which are the primary power sources in front-end circuits. Therefore, quantized precoding algorithms were recently proposed to address this issue. They used low-resolution DACs at the transmitter and compensated for the distortion caused by the quantization effect at the baseband. This paper proposes a constellation-range gain-controlled lattice-reduction-aided (CR-GCLRA) one-bit precoding algorithm for massive MU-MIMO systems with high-order quadrature amplitude modulation (QAM) signaling. Lattice reduction (LR) preprocessing is adopted for performance enhancement. A novel gain-control mechanism is proposed to address the constellation range expansion problem in conventional LR preprocessing. We designed and implemented a CR-GCLRA one-bit precoding processor by using TSMC 40-nm CMOS technology to accelerate the one-bit precoding processing for 64-QAM $4 \times 64$ MU-MIMO system. The proposed one-bit precoding processor can achieve a throughput of 43.92 Mbits per second at a clock speed of 269 MHz with 271 mW power consumption and $0.51\mu \text{s}$ latency.

Massive multi-user multiple-input multiple-output (MU-MIMO) communication is a crucial technique for next-generation wireless systems because it enables high-throughput and ultra-reliable data transmission. However, high power consumption is a challenging problem in conventional transceiver designs, where each RF chain at the transmitter requires a pair of high-resolution digital-to-analog converters (DACs), which are the primary power sources in front-end circuits. Therefore, quantized precoding algorithms were recently proposed to address this issue. They used low-resolution DACs at the transmitter and compensated for the distortion caused by the quantization effect at the baseband. This paper proposes a constellation-range gain-controlled lattice-reduction-aided (CR-GCLRA) one-bit precoding algorithm for massive MU-MIMO systems with high-order quadrature amplitude modulation (QAM) signaling. Lattice reduction (LR) preprocessing is adopted for performance enhancement. A novel gain-control mechanism is proposed to address the constellation range expansion problem in conventional LR preprocessing. We designed and implemented a CR-GCLRA one-bit precoding processor by using TSMC 40-nm CMOS technology to accelerate the one-bit precoding processing for 64-QAM $4 \times 64$ MU-MIMO system. The proposed one-bit precoding processor can achieve a throughput of 43.92 Mbits per second at a clock speed of 269 MHz with 271 mW power consumption and $0.51\mu \text{s}$ latency.

Here we consider the question whether the lattice reformulation of a linear integer program can be used to produce effective cutting planes. In particular, we aim at deriving split cuts that cut off more of the integrality gap than Gomory mixed-integer (GMI) inequalities generated from LP-tableaus, while being less computationally demanding than generating the split closure. We consider integer programs (IPs) in the form max{ Ax=b x =Zn+}, where the reformulation takes the form max\{cx +cQ> -xu u =Zn-m Z n - m\}, where Q is an n (n - m) integer matrix. Working on an optimal LP-tableau in the u -space allows us to generate n - m GMIs in addition to the m GMIs associated with the optimal tableau in the x space. These provide new cuts that can be seen as GMIs associated to n - m nonelementary split directions associated with the reformulation matrix \Q . On the other hand it turns out that the corner polyhedra associated to an LP basis and the GMI or split closures are the same whether working in the x or u spaces. Our theoretical derivations are accompanied by an illustrative computational study. The computations show that the effectiveness of the cuts generated by this approach depends on the quality of the reformulation obtained by the reduced basis algorithm used to generate Q and that it is worthwhile to generate several rounds of such cuts. However, the effectiveness of the cuts deteriorates as the number of constraints is increased.

The mine safety performance evaluation and its law were taken as research objects; the human factor, mechanical equipment, management system, environment and technology were taken as mine safety state indexes; accident loss and disabling incident frequency rate as mine safety performance evaluation indexes. The classification standard of safety performance was determined according to the range of influence index and evaluation index, and then the influence index value of safety performance was discretized. According to the classification standard, the classical domain and node domain of safety performance evaluation index classification were determined. Combined with the weight of each safety performance evaluation index, the correlation degree of each mine safety performance evaluation was calculated, and then the mine safety performance was evaluated, and the mine safety performance evaluation decision system was constructed. Based on the dominance-based rough set, the evaluation law of mine safety performance evaluation decision system was analyzed. To generate more reliable laws, an efficient concept lattice reduction method was introduced to reduce the attributes of decision system. The example shows that when the concept lattice is used for reduction, there are 84 concepts in the Hasse diagram, and 24 of them can be identified. The decision system has 8 conditional attributes. And 255 power sets can be obtained under 8 conditional attributes. After eliminating the defect attribute, there are 148 reducible sets, among which there are 4 optimal reducible sets with two conditional attributes. The results show that the proposed method can be used to analyze the law of mine safety performance, so as to improve mine safety work.

Lattice reduction (LR) techniques have been widely adopted to enhance the detection performance with low additional complexity in spatial multiplexed multiple-input multiple-output (MIMO) systems. However, in large MIMO systems, the required number of iterations and the corresponding latency and complexity of the existing LR algorithms increase exponentially. In this brief, we propose a low-latency LR method based on Seysen’s algorithm (SA) for high-dimensional MIMO systems. The proposed algorithm introduces dual-index selection and simultaneously updates two basis pairs without the recalculation of metrics to reduce the latency and complexity. Additionally, it introduces a reduced candidate set to decrease the computational complexity and memory usage. The numerical results demonstrate that the proposed LR-aided linear detector outperforms the conventional Lenstra-Lenstra-Lovasz algorithm. Furthermore, the performance of the proposed method is comparable to that of SA; it achieves reductions of 40% and 12.5% in the number of iterations and computational complexity, respectively. The proposed 65-nm CMOS implementation achieves an overall latency of 196 ns and a throughput of 25.5 M matrix/s with a clock frequency of 357.1 MHz for $8 \times 8$ MIMO systems.

Due to the continuous demand for higher bit rates, the management of the spatio-temporal intersymbol interference in frequency-selective multiple-input multiple-output (MIMO) channels becomes increasingly important. For single-input single-output channels, equalized precoded partial-response signaling is capable of handling a large amount of intersymbol interference, but, to date, no equalization scheme with general partial-response signaling has been presented for the frequency-selective MIMO channel. Not only does this contribution extend partial-response signaling to the MIMO channel by proposing a general spatio-temporal partial-response precoder, but it also develops a minimum mean-squared-error optimization framework in which the equalization coefficients and the spatio-temporal target response are jointly optimized. Three iterative optimization algorithms are discussed, which update (part of) a row of the target impulse response matrix in each iteration. In particular, the third algorithm reformulates this row optimization as a lattice decoding problem. Numerical simulations confirm that the general partial-response signaling clearly outperforms the traditional full-response signaling in terms of the mean squared error and the bit error rate. The third optimization algorithm has a better performance but a higher complexity, compared to the first and the second algorithm.

In recent years, the block Korkine-Zolotarev (BKZ) and its variants such as BKZ 2.0 have been used as de facto algorithms to estimate the security of a lattice-based cryptosystem. In 2017, DeepBKZ was proposed as a mathematical improvement of BKZ, which calls LLL with deep insertions (DeepLLL) as a subroutine alternative to LLL. DeepBKZ can find a short lattice vector by smaller blocksizes than BKZ. In this paper, we develop a self-dual variant of DeepBKZ, as in the work of Micciancio and Walter for self-dual BKZ. Like DeepBKZ, our self-dual DeepBKZ calls both DeepLLL and its dual variant as main subroutines in order to accelerate to find a very short lattice vector. We also report experimental results of DeepBKZ and our self-dual DeepBKZ for random bases on the Darmstadt SVP challenge.

Lattice basis reduction is a mandatory tool for solving lattice problems such as the shortest vector problem. The Lenstra–Lenstra–Lovasz reduction algorithm (LLL) is the most famous, and its typical improvements are the block Korkine–Zolotarev algorithm and LLL with deep insertions (DeepLLL), both proposed by Schnorr and Euchner. In BKZ with blocksize $$\beta $$ , LLL is called many times to reduce a lattice basis before enumeration to find a shortest non-zero vector in every block lattice of dimension $$\beta $$ . Recently, “DeepBKZ” was proposed as a mathematical improvement of BKZ, in which DeepLLL is called as a subroutine alternative to LLL. In this paper, we analyze the output quality of DeepBKZ in both theory and practice. Specifically, we give provable upper bounds specific to DeepBKZ. We also develop “DeepBKZ 2.0”, an improvement of DeepBKZ like BKZ 2.0, and show experimental results that it finds shorter lattice vectors than BKZ 2.0 in practice.

Under the threat of quantum computers’ expected powerful computational capacity, the study on post-quantum cryptography is becoming urgent nowadays. Lattice-based cryptography is one of the most promising candidates of post-quantum cryptography. To give a secure instantiation for practical applications, it is necessary to understand the complexity of the best-known attacks. Most of the attacks to lattice-based cryptography use basis reduction algorithms. For instance, the most commonly used practical basis reduction algorithms are variants of the block Korkin–Zolotarev (BKZ) algorithm. In this paper, we study the effect of applying the quick reordering technique (QRT) to lattice algorithms, mainly the enumeration algorithm and the BKZ algorithm. We show that QRT is a simple method to improve these two algorithms with respect to cutting down the number of search nodes and thus reducing the total runtime. For improving on the LLL algorithm with dimensions smaller than 30, the success rate is larger than 10%, and for the BKZ algorithm with blocksize smaller than 30, the success rate is larger than 40%. At first, we observe that reordering the LLL-reduced basis vectors by increasing norm orders will change the distribution of search nodes in the enumeration tree, which gives a chance to reduce the enumeration search nodes with a certain probability. The experimental results show that the runtime of the enumeration algorithm can be accelerated approximately by a factor of two. We further explain this phenomenon from a theoretical point of view, which follows Gama–Nguyen–Regev’s analysis (Gama et al., in: Advances in cryptology—EUROCRYPT 2010, proceedings of 29th annual international conference on the theory and applications of cryptographic techniques, pp 257–278, 2010). Then we apply this reordering technique to the implementation of the BKZ algorithm in the open-source library NTL. Our experimental results in dimensions 100–120 with blocksize 15–30 show that on the LLL-reduced bases, our modified NTL-BKZ outputs a vector shorter than the original NTL-BKZ with rate 40.91%-45.73% by setting the LLL approximation factor by $$\delta _{LLL}=0.99$$ . Furthermore, in the instances where the improved BKZ found one same or shorter vector, the runtime is up to 2.02 times faster than the original NTL-BKZ when setting the blocksize $$\beta =25$$ with $$\delta _{LLL}=0.99$$ .

Multiple-symbol differential detection MSDD algorithms are proposed in two-way relay transmission TWRT . Firstly, generalized likelihood ratio test based MSDD GLRT-MSDD is proposed in TWRT. Unfortunately, as the number of observation windows increases, the computational complexity of GLRT-MSDD increases exponentially. Hence, this detection in TWRT constitutes a challenging problem. Moreover, we find a way to reformulate the GLRTMSDD model and additionally propose a lattice-reduction aided MSDD LR-MSDD model. Performance analysis and simulations show that the proposed LR-MSDD provides bit-error rate performance close to that of GLRT-MSDD with lower complexity in TWRT.

Pseudorandom sequences are widely used in cryptography. Multiple recursive generators are an important class of pseudorandom sequence generators. A typical application is to obtain truncated sequences by truncating partial bits of the sequences output by the generators. This paper studies the predictability of truncated multiple recursive generators with unknown parameters. Given a few truncated digits of high-order bits output by a multiple recursive generator, we give a method based on lattice reduction to recover the parameters and the initial state of the generator. Our method is an extension of Stern’s algorithm which was proposed to predict the truncated sequences of linear congruential generators.

Let [Formula: see text] be an RSA public key with private exponent [Formula: see text] where [Formula: see text] and [Formula: see text] are large primes of the same bit size. At Eurocrypt 96, Coppersmith presented a polynomial-time algorithm for finding small roots of univariate modular equations based on lattice reduction and then succussed to factorize the RSA modulus. Since then, a series of attacks on the key equation [Formula: see text] of RSA have been presented. In this paper, we show that many of such attacks can be unified in a single attack using a new notion called Coppersmith’s interval. We determine a Coppersmith’s interval for a given RSA public key [Formula: see text] The interval is valid for any variant of RSA, such as Multi-Prime RSA, that uses the key equation. Then we show that RSA is insecure if [Formula: see text] provided that we have approximation [Formula: see text] of [Formula: see text] with [Formula: see text] [Formula: see text] The attack is an extension of Coppersmith’s result.

This paper proposes a high-efficient preprocessing algorithm for $16\times 16$ MIMO detections. The proposed algorithm combines a sorting-relaxed QR decomposition (SRQRD) and a modified greedy LLL (MGLLL) algorithm. First, SRQRD is conducted to decompose the channel matrices. This decomposition adopts a relaxed sorting strategy together with a paralleled Givens Rotation (GR) array scheme, which can reduce the processing latency by 60% compared with conventional sorted QR decomposition (SQRD). Then, an MGLLL algorithm is conducted to improve detection performance further. The MGLLL algorithm adopts a paralleled selection criterion, and only process the most urgent iterations. Thus the processing latency and column swaps can be reduced by 50% and 75%, respectively, compared with the standard LLL algorithm. Finally, the bit-error-rate (BER) performance of this preprocessing algorithm is evaluated using two MIMO detectors. Results indicate that this preprocessor suffers a negligible performance degradation compared with the combination of the standard LLL algorithm and SQRD. Based on this preprocessing algorithm, a pipelined hardware architecture is also designed in this paper. A series of systolic coordinated-rotation-digital-computer (CORDIC) arrays are utilized, and highly-pipelined circuits are designed, helping this architecture achieve high frequency performance. This architecture is implemented using 65-nm CMOS technology, which can work at a maximum frequency of 625 MHz to process channel matrices every 16 clock cycles. The latency is 0.9 us. Comparisons indicate that this preprocessor outperforms other similar designs in terms of latency, throughput, and gate-efficiency.

As a non-linear precoding alternative to Tomlinson-Harashima precoding (THP), in this paper, so-called lattice reduction aided precoding (LRP) is considered as a crosstalk precompensation technique for downstream transmission in G.fast DSL networks. First, a practically achievable bit-rate expression for LRP is proposed in function of the precoder and integer matrix. The problem then consists of a joint precoder and integer matrix design in order to maximize the weighted sum-rate (WSR) under per-line power constraints. For a fixed integer matrix, zero-forcing (ZF) precoder matrix design simplifies to gain scaling optimization with complex gain scalars, for which a successive lower bound maximization method is presented. Additionally, it is established that the achievable ZF-LRP sum-rate is upper bounded by the achievable ZF-THP sum-rate at high SNR. For computing the optimal precoder matrix, on the other hand, an efficient method is developed by leveraging on the equivalence between the WSR maximization and the weighted sum of mean squared error (MSE) minimization, leading to a locally-optimal MMSE-LRP solution. Simulations with a measured G.fast cable binder are provided to compare the proposed LRP schemes with THP schemes.

In MIMO OFDM (Multiple Input Multiple Output Orthogonal Frequency Division Multiplexing) cellular systems with frequency reuse factor of unity, the cell edge users experience interference termed as Inter Cell Interference (ICI). ICI degrades the throughput of the cell, when many users exist at cell edges. ICI can be minimized by proper optimal decoding techniques like Maximum Likelihood Decoding (MLD) which is too complex and not possible to implement practically. Suboptimal decoders like sphere decoder is used to achieve the performance near to MLD with reduced complexity. The sphere decoder begins the search process by choosing a proper initial search radius. The paper proposes an optimal radius choice algorithm, to determine the initial search radius for sphere decoder based on system specifications like size of MIMO, modulation technique, Signal to Noise Ratio statistics. A Look Up Table is prepared based on these statistics. A combination of radius selection and radius updation with pre processing is considered. The pre processing methods named Lattice Reduction is used to convert Integer Least Square problem to simple polynomial. Radius is updated using Probabilistic Tree Pruning or Increasing Radius Algorithm. Simulations demonstrate that Increasing Radius Algorithm performs better at larger SNR regions i.e. above 5 dB and Reducing Radius Search algorithm at lower SNR. The initial radius considered from LUT is very efficient selection of search radius. It leads to reduced difference, between final radius and selected radius. This difference is less than 10% for both IRS-SD and RRS-SD, without degrading the performance.

Lattices are a popular field of study in mathematical research, but also in more practical areas like cryptology or multiple-input/multiple-output (MIMO) transmission. In mathematical theory, most often lattices over real numbers are considered. However, in communications, complex-valued processing is usually of interest. Besides, by the use of dual-polarized transmission as well as by the combination of two time slots or frequencies, four-dimensional (quaternion-valued) approaches become more and more important. Hence, to account for this fact, well-known lattice algorithms and related concepts are generalized in this work. To this end, a brief review of complex arithmetic, including the sets of Gaussian and Eisenstein integers, and an introduction to quaternion-valued numbers, including the sets of Lipschitz and Hurwitz integers, are given. On that basis, generalized variants of two important algorithms are derived: first, of the polynomial-time LLL algorithm, resulting in a reduced basis of a lattice by performing a special variant of the Euclidean algorithm defined for matrices, and second, of an algorithm to calculate the successive minima - the norms of the shortest independent vectors of a lattice - and its related lattice points. Generalized bounds for the quality of the particular results are established and the asymptotic complexities of the algorithms are assessed. These findings are extensively compared to conventional real-valued processing. It is shown that the generalized approaches outperform their real-valued counterparts in complexity and/or quality aspects. Moreover, the application of the generalized algorithms to MIMO communications is studied, particularly in the field of lattice-reduction-aided and integer-forcing equalization.

Howgrave-Grahams Hybrid-Attacke wird als grundlegende Technik genutzt, um die Sicherheit des NTRU Public-Key-Kryptosystems zu evaluieren. Daher spielt dieser Angriff bei der Entwicklung des NTRU Kryptographie-Schemas, welches auch bei NIST, als Post-Quanten- Kryptosystem für die potentielle Nachfolge aktueller Public-Key-Kryptosysteme eingereicht wurde eine wichtige Rolle. Bevor diese Art von Angriff auf NTRU etabliert wurde, gab es Ansätze, NTRU mit der Meetin- the-Middle-Methode oder auf der Basis von Gitterreduktion anzugreifen. Diese Methoden haben insofern wesentlich zur Entstehung der Hybrid-Attacke beigetragen, da sie bei dieser entsprechend modifiziert kombiniert werden. In dieser Bachelorarbeit werden die mathematischen Techniken, die im Zuge der Meet-in-the- Middle-Methode und des Angriffs mit Gitterreduktion angewandt werden näher betrachtet. Weiterführend wird die Art und Weise analysiert, wie diese beiden Angriffe in der Hybrid- Attacke kombiniert werden, um einen effizienteren Angriff auf NTRU zu vollziehen. Obwohl durch die Hybrid-Attacke die Sicherheit von NTRU nicht gebrochen wird, bietet diese eine wesentliche Methodik um das NTRU System und neue Möglichkeiten von NTRU Parameter zu prüfen. Howgrave-Grahams Hybrid Attack is still handled as a fundamental technique challenging the hardness of the NTRUEncrypt public key cryptosystem. Hence, the Hybrid Attack plays a vital role considering the development of a post-quantum cryptography scheme, which was also submitted to NIST [1] for being a suitable successor of today’s public key cryptosystems. There have been approaches to attack NTRU before with the Meet-in-the-Middle method or attacks based on lattice reduction, and these two types of attacks are crucial as they provide the basis of the Hybrid Attack. In this bachelor thesis, we take a closer look at the mathematical mechanisms applied in the Meet-in-the-Middle and lattice reduction attack to reconstruct how these attacks address a not directly lattice-based cryptography scheme, like the NTRU. Further, we reflect upon how these two attacks are combined to create a more powerful and efficient attack, which provides - although not breaking NTRU - an essential method to test, estimate and evaluate the security of new parameter sets of NTRU.

Plane-probing algorithms have become fundamental tools to locally capture arithmetical and geometrical properties of digital surfaces (boundaries of a connected set of voxels), and especially normal vector information. On a digital plane, the overall idea is to consider a local pattern, a triangle, that is expanded starting from a point of interest using simple probes of the digital plane with a predicate "Is a point x in the digital plane?". Challenges in plane-probing methods are to design an algorithm that terminates on a triangle with several geometrical properties: its normal vector should match with the expected one for digital plane (correctness), the triangle should be as compact as possible (acute or right angles only), and probes should be as close as possible to the source point (locality property). In addition, we also wish to minimize the number of iterations or probes during the computations. Existing methods provide correct outputs but only experimental evidence for these properties. In this paper, we present a new plane-probing algorithm that is theoretically correct on digital planes, and with better experimental compactness and locality than existing solutions. Additional properties of this new approach also suggest that theoretical proofs of the aforementioned geometrical properties could be achieved.

Die Schwierigkeit die Nullstellen von Polynomen im Ring $\mathbb{Z}_N$ zu finden, ist eine der zugrundeliegenden Problemstellungen des RSA Verfahrens. Dieses Problem wurde in den 90er Jahren von Don Coppersmith aufgegriffen und er beeinflusste damit die Forschung zur Sicherheit von RSA grundlegend. Er entwickelte ein Verfahren, welches es in speziellen Fällen ermöglicht mit Hilfe von Gitterreduktion und insbesondere des LLL-Algorithmus von Lenstra Lenstra and Lovász die verschlüsstelte Nachricht zu rekonstruieren.\\ Diese Bachelorarbeit bietet einen Überblick über die mathematischen Hintergründe der Coppersmith Methode und stellt den Bezug zu zwei vereinfachten RSA Problemen her. Obwohl RSA und die heute üblichen Implementierungen von diesen Ansätzen nicht ernsthaft gefährdet werden, bekommen wir einen Einblick wie vereinfachte RSA Probleme mathematisch gelöst werden können. The hardness of the root-finding problem of univariate modular polynomials is essential for ensuring the security of the RSA cryptosystem. However, in the late 1990s, Don Coppersmith constructed an approach that addressed this problem and took the research concerning RSA security to another level. By using lattice reduction, especially the LLL-Algorithm of Lenstra Lenstra and Lovász, the reconstruction of the plain text message is, given relaxed scenarios, possible.\\ This thesis provides an overview of the mathematical background of the Coppersmith Method and draws the line to its application on two relaxed RSA problems: Stereotyped message and related RSA messages. As a result, we see a relation between those problems, even given different problem situations.\\ Although RSA and today's RSA implementations do not seriously get threatened by the examined approaches, we get an insight into the possibility to solve relaxed RSA problems mathematically with a method, which researchers still continue to improve.

The modulus of type N=p2q is often used in many variants of factoring-based cryptosystems due to its ability to fasten the decryption process. Faster decryption is suitable for securing small devices in the Internet of Things (IoT) environment or securing fast-forwarding encryption services used in mobile applications. Taking this into account, the security analysis of such modulus is indeed paramount. This paper presents two cryptanalyses that use new enabling conditions to factor the modulus N=p2q of the factoring-based cryptosystem. The first cryptanalysis considers a single user with a public key pair (e,N) related via an arbitrary relation to equation er−(Ns+t)=αp2+βq2, where r,s,t are unknown parameters. The second cryptanalysis considers two distinct cases in the situation of k-users (i.e., multiple users) for k≥2, given the instances of (Ni,ei) where i=1,…,k. By using the lattice basis reduction algorithm for solving simultaneous Diophantine approximation, the k-instances of (Ni,ei) can be successfully factored in polynomial time.

We revisit the problem of integer factorization with number-theoretic oracles, including a well-known problem: can we factor an integer $N$ unconditionally, in deterministic polynomial time, given the value of the Euler totient $ϕ(N)$? We show that this can be done, under certain size conditions on the prime factors of N. The key technique is lattice basis reduction using the LLL algorithm. Among our results, we show for example that if $N$ is a squarefree integer with a prime factor $p > √ N$ , then we can recover p in deterministic polynomial time given $ϕ(N)$. We also shed some light on the analogous problems for Carmichael's function, and the order oracle that is used in Shor's quantum factoring algorithm.

Conference Name: 2021 IEEE International Black Sea Conference on Communications and Networking (BlackSeaCom) Date of Conference: 24-28 May 2021 We consider the downlink of a multiuser multiple-input multiple-output (MU-MIMO) system, where each user feeds back a partial channel state information (CSI), namely, the quantized version of the dominant eigenvector of its channel covariance matrix, to the base station (BS) for precoding. Specifically, we propose a downlink multiuser precoding scheme by first reconstructing the equivalent channel matrix of each user via a limited feedback, and then by employing a precoder to suppress the multiuser interference at the receivers. For the single stream case, a signal-to-leakage-and-noise ratio (SLNR) based precoding is employed, while for the full stream case with limited feedback, we employ a lattice reduction aided block diagonalization type precoding with suitable modifications at the receiver side. Extensive numerical examples which are provided using the 5G new radio (5G-NR) channel models demonstrate that the proposed schemes outperform the existing eigenvector based algorithms, and they are more robust against the downlink channel estimation errors.

The lll algorithm is a polynomial-time algorithm for reducing d-dimensional lattice with exponential approximation factor. Currently, the most efficient variant of lll, by Neumaier and Stehle, has a theoretical running time in \(d^4\cdot B^{1+o\left( 1\right) }\) where B is the bitlength of the entries, but has never been implemented. This work introduces new asymptotically fast, parallel, yet heuristic, reduction algorithms with their optimized implementations. Our algorithms are recursive and fully exploit fast matrix multiplication. We experimentally demonstrate that by carefully controlling the floating-point precision during the recursion steps, we can reduce euclidean lattices of rank d in time \(\tilde{O}(d^\omega \cdot C)\), i.e., almost a constant number of matrix multiplications, where \(\omega \) is the exponent of matrix multiplication and C is the log of the condition number of the matrix. For cryptographic applications, C is close to B, while it can be up to d times larger in the worst case. It improves the running-time of the state-of-the-art implementation fplll by a multiplicative factor of order \(d^2\cdot B\). Further, we show that we can reduce structured lattices, the so-called knapsack lattices, in time \(\tilde{O}(d^{\omega -1}\cdot C)\) with a progressive reduction strategy. Besides allowing reducing huge lattices, our implementation can break several instances of Fully Homomorphic Encryption schemes based on large integers in dimension 2,230 with 4 millions of bits.

The public parameters of the RSA cryptosystem are represented by the pair of integers N and e . In this work, first we show that if e satisfies the Diophantine equation of the form e x 2 − ϕ ( N ) y 2 = z for appropriate values of x , y and z under certain specified conditions, then one is able to factor N . That is, the unknown y x can be found amongst the convergents of e N via continued fractions algorithm. Consequently, Coppersmith’s theorem is applied to solve for prime factors p and q in polynomial time. We also report a second weakness that enabled us to factor k instances of RSA moduli simultaneously from the given ( N i , e i ) for i = 1 , 2 , ⋯ , k and a fixed x that fulfills the Diophantine equation e i x 2 − y i 2 ϕ ( N i ) = z i . This weakness was identified by solving the simultaneous Diophantine approximations using the lattice basis reduction technique. We note that this work extends the bound of insecure RSA decryption exponents.

Recently, the calculus of Formal Concept Analysis (FCA) in the fuzzy setting is prolonged in bipolar fuzzy space for precise analysis of fuzzy attributes. In this process it is addressed that the attributes like bald (or tadpole) or not bald (not tadpole) cannot be defined through a precise or sharp boundaries. To deal with them, some evidence to support (i.e. true t A membership-values) or reject (i.e. false f A membership-values) the attributes is required in the given boundary 0 ≤ t A + f A ≤ 1 . Hence, the proposed method tried to provide the mathematical algebra of vague concept lattice and its navigation at user defined granules with an illustrative example. In addition, the vague entropy measurement is also computed to validate the results.

We present a technique to enhance the security of the Goldreich, Goldwasser and Halevi (GGH) scheme. The security of GGH has practically been broken by lattice reduction techniques. Those attacks are successful due to the structure of the basis used in the secret key. In this work, we aim to present a new technique to alleviate this problem by modifying the public key which hides the structure of the corresponding private key. We intersect the initial lattice with a random one while keeping the initial lattice as our secret key and use the corresponding result of the intersection as the public key. We show sufficient evidence that this technique will make GGH implementations secure against the aforementioned attacks.

This paper presents a new low complexity sphere decoding (SD) for the generalized spatial modulation (GSM). We introduce a pre-processing stage using the lattice reduction (LR) aided minimum mean squared error (MMSE) equalization in the GSM systems. This stage speeds up the search in the decoding tree and provides a lattice dependent (LD) initial choice of the radius. Moreover, we derive a lattice independent (LI) initial radius that guarantees the optimal performance at a high signal-to-noise ratio (SNR). We also propose an iterative method to increase the radius in order to achieve the maximum likelihood (ML) performance at all SNRs. We show that the proposed algorithm achieves the ML performance while requiring prominently less computational complexity (CC) than an exhaustive search. In addition, we analyze the CC of the resulting algorithm at high SNRs and we derive an analytical expression for the complexity. The simulation results demonstrate a noticeable decrease in the CC of the proposed sphere decoding algorithm in comparison with its counterparts, particularly at low SNRs.

Sampling from the lattice Gaussian distribution has emerged as an important problem in coding, decoding, and cryptography. In this paper, lattice reduction technique is adopted to Gibbs sampler for lattice Gaussian sampling. First, with respect to lattice Gaussian distribution, we show the convergence rate of systematic scan Gibbs sampling is characterized by the Hirschfeld-Gebelein-Renyi maximal correlation among the multivariate of being sampled. Then, Lattice-reduction-aided Gibbs algorithm is proposed to sample from an equivalent lattice Gaussian distribution but with less correlated multivariate, thus leading to a better Markov mixing. After that, we extend the proposed lattice-reduction-aided Gibbs sampling to lattice decoding, where the choice of the standard deviation for the sampling is fully investigated. A customized solution that suits for each specific decoding case by Euclidean distance is given, which results in a better tradeoff between Markov mixing and sampler decoding. Based on it, a startup mechanism is also proposed for Gibbs sampler decoding, where decoding complexity can be reduced without performance loss. Moreover, the recycling Gibbs sampling that exploits the potential of samples is also considered to improve the decoding performance in lattice decoding. Simulation results based on large-scale uncoded multiple-input multiple-output detection are presented to confirm the performance gain and complexity reduction.

In this paper, a very low complexity Lattice Reduction technique, called Dual Shortest Longest Vector algorithm (SLV), is adopted to improve the Bit Error Rate (BER) performance of the Minimum Mean Square Error (MMSE) detector in high-load Massive MIMO systems, whereby resulting in the so-called SLV-aided MMSE (MMSE–SLV) detector. An efficient combination scheme of Generalized Group Detection (GGD) algorithm and the MMSE–SLV, called MMSE–GGD–SLV, is further proposed to enhance BER performance of the system more significantly. In order to do so, we first convert the Group Detection approach to the generalized one (GGD) by creating an arbitrary number of sub-systems. Then, an additional operation, i.e., channel matrix sorting, is applied to the GGD to reduce the error propagation between sub-systems. To make the detection complexities of the MMSE–GGD–SLV detector more practical, the MMSE–SLV detection procedure is only applied to the first sub-system. Various BER performance simulations and complexity analysis show that both the MMSE–GGD–SLV and the MMSE–SLV detectors noticeably outperform their conventional MMSE counterpart, yet at the cost of higher detection complexities. However, their complexities are kept at acceptable levels, which are much lower than those of the conventional BLAST detector. Therefore, the proposed detectors are very good candidates for signal recovery in high load Massive MIMO systems.

One-bit precoding is a recently proposed scheme that facilitates the use of one-bit digital-to-analog converters to reduce the cost and power consumption in a massive antenna system. In line with this, we consider the massive multiuser-multiple-input multiple-output scenario and propose a lattice-reduction-aided one-bit precoding algorithm. The proposed algorithm solves the mean square error-based one-bit precoder design problem by augmenting the Babai's nearest plane procedure with a newly proposed list generation algorithm. The proposed precoder contains no tuning parameters and hence avoids additional overheads for parameter tuning when applied to a practical system. Furthermore, the symbol precoding stage in our precoder is noniterative and can be implemented via distributed processing. These desirable features along with an excellent performance-complexity tradeoff demonstrated that the proposed precoder stands out as a competitive alternative in addition to the existing one-bit precoders.

Lattice basis reduction algorithms have been used in cryptanalysis. The most famous algorithm is LLL, proposed by Lenstra, Lenstra, Lovasz, and one of its typical improvements is LLL with deep insertions (DeepLLL). A DeepLLL-reduced basis is LLL-reduced, and hence its quality is at least as good as LLL. In practice, DeepLLL often outputs a more reduced basis than LLL, but no theoretical result is known. First, we show provable output quality of DeepLLL, strictly better than that of LLL. Second, as a main work of this paper, we propose a new variant of DeepLLL. The squared-sum of Gram–Schmidt lengths of a basis is related with the computational hardness of lattice problems such as the shortest vector problem (SVP). Given an input basis, our variant monotonically decreases the squared-sum by a given factor at every deep insertion. This guarantees that our variant runs in polynomial-time.

Generalized Frequency Division Multiplexing (GFDM) is a promising candidate waveform for the air interface of the Fifth Generation (5G) communication networks. The flexibility of GFDM enables it to meet different application requirements of future networks. Nevertheless, the combination of GFDM and Multiple-Input Multiple-Output (MIMO) systems using Spatial Multiplexing (SM) is a necessity to achieve high spectral efficiency. However, spatial demultiplexing using common SM detection algorithms becomes an even more challenging task due to the inherent self-interference of GFDM. In this paper, we investigate a class of Lattice-Reduction-Aided (LRA) signal detection algorithms that can achieve an attractive tradeoff between performance and complexity in MIMO–GFDM systems. Simulation results show that with less increase in complexity, LRA algorithms can greatly improve the performance of the classical linear detector.