Vector Problem Research Articles

A physical layer security scheme for 6G wireless networks using post-quantum cryptography

The sixth generation (6G) of mobile networks is poised to revolutionize communication capabilities with its infinite-like reach. These networks will feature an ultra-dense topology, accommodating a wide range of devices, from macro devices like satellites to nano-devices integrated within the human body. However, the extensive data traffic handled by 6G networks, a significant portion of which is sensitive in nature, presents a security challenge. This paper presents the design and implementation of an encryption scheme that ensures the confidentiality of data transmission in the physical layer of 6G networks. The proposed physical layer security architecture relies on lattice cryptography, wherein each user is associated with a pair of bases (a public basis and a private basis) and a set of orthogonal sub-carriers. The encryption process involves projecting the vector of the transmitted data’s quadrature amplitude modulation (QAM) symbols onto the user’s public basis. A random error vector is then added before applying the inverse Fourier transform of the orthogonal frequency division multiplexing (OFDM) technique. The security of this encryption scheme hinges on the complexity of the closest vector problem in an integer lattice. To evaluate its efficacy, we analyze the security of our lattice-based physical layer encryption concerning the properties of the base pairs and error vectors. Our findings indicate that the proposed scheme offers satisfactory security protection against eavesdropping attacks by significantly enhancing the confidentiality of the transmitted signal. Furthermore, we assess the performance of our design through numerical experiments, demonstrating its resilience against various security attacks.

Concrete analysis of approximate ideal-SIVP to decision ring-LWE reduction

<p style='text-indent:20px;'>A seminal 2013 paper by Lyubashevsky, Peikert, and Regev proposed basing post-quantum cryptography on ideal lattices and supported this proposal by giving a polynomial-time security reduction from the approximate Shortest Independent Vectors Problem (SIVP) to the Decision Learning With Errors (DLWE) problem in ideal lattices. We give a concrete analysis of this multi-step reduction. We find that the tightness gap in the reduction is so great as to vitiate any meaningful security guarantee, and we find reasons to doubt the feasibility in the foreseeable future of the quantum part of the reduction. In addition, when we make the reduction concrete it appears that the approximation factor in the SIVP problem is far larger than expected, a circumstance that causes the corresponding approximate-SIVP problem most likely not to be hard for proposed cryptosystem parameters. We also discuss implications for systems such as Kyber and SABER that are based on module-DLWE.</p>

A Novel Approximation Algorithm for the Shortest Vector Problem

A Novel Approximation Algorithm for the Shortest Vector Problem

Digital Transformation of Optimal Decision-making in Economic and Engineering Systems Based on the Theory and Methods of Vector Optimization

Objective: The purpose of the work is to analyze, study the processes of digital transformation and make optimal decisions in economic and engineering systems based on the theories and methods of vector optimization. Methods: Methods for studying the processes of digital transformation of economic and engineering systems are based on the author's theory of vector optimization, which is represented by: axiomatics and optimality principles that show how one solution is better than another. On the basis of the principles of optimality, constructive methods for solving vector problems have been built, firstly, with equivalent criteria, and, secondly, with a given priority of the criterion: https://rdcu.be/bhZ8i (Springer Nature's "Vector optimization with equivalent and priority criteria" is distributed free of charge.) The paper presents that the problem of accepting the optimal process is carried out as a result of the digital transformation of the initial (statistical, experimental) data into vector problems in optimization, which represents a mathematical model of economic and engineering systems. Depending on the number and type of criteria, we solve problems with a vector of criteria, each component of which is directed to the maximum, and with a vector of criteria, each component of which is aimed at a minimum, in aggregate. Results: The result of the study is the practical orientation of digital transformation, which is shown in the computer-aided design of economic and engineering systems based on vector optimization. For this purpose, software for solving vector problems of linear and nonlinear programming has been developed. Linear vector problems are used in economic systems (forecasting, planning). Software for solving vector problems of nonlinear programming are used digital transformation of optimal decision-making in engineering problems. Conclusion: In conclusion, numerical examples in the field of economic systems are presented by the digital transformation of optimal decision-making in the development of the company. When making optimal decisions in engineering systems, the following has been developed: the construction of initial data (technical task) for modeling; transformation of a mathematical model under uncertainty into a model under conditions of certainty; making an optimal decision with equivalent criteria; making the optimal decision with a given priority criterion

Algorithm unfolding for block-sparse and MMV problems with reduced training overhead

In this study, we consider algorithm unfolding for the multiple measurement vector (MMV) problem in the case where only few training samples are available. Algorithm unfolding has been shown to empirically speed-up in a data-driven way the convergence of various classical iterative algorithms, but for supervised learning, it is important to achieve this with minimal training data. For this, we consider learned block iterative shrinkage thresholding algorithm (LBISTA) under different training strategies. To approach almost data-free optimization at minimal training overhead, the number of trainable parameters for algorithm unfolding has to be substantially reduced. We therefore explicitly propose a reduced-size network architecture based on the Kronecker structure imposed by the MMV observation model and present the corresponding theory in this context. To ensure proper generalization, we then extend the analytic weight approach by Liu and Chen to LBISTA and the MMV setting. Rigorous theoretical guarantees and convergence results are stated for this case. We show that the network weights can be computed by solving an explicit equation at the reduced MMV dimensions which also admits a closed-form solution. Toward more practical problems, we then considered convolutional observation models and show that the proposed architecture and the analytical weight computation can be further simplified and thus open new directions for convolutional neural networks. Finally, we evaluate the unfolded algorithms in numerical experiments and discuss connections to other sparse recovering algorithms.

On fluxes in the 19 Landau-Ginzburg model

In this paper we present a large class of flux backgrounds and solve the shortest vector problem in type IIB string theory on an orientifold of the 19 Landau-Ginzburg model.

Solvability of the Quaternary Continuous Classical Boundary Optimal Control Dominated by Quaternary Parabolic System

The purpose of this paper is to study the solvability of the quaternary continuous classical boundary optimal control vector problem dominated by quaternary nonlinear parabolic boundary value problem with state constraints. The existence theorem for a quaternary continuous classical boundary optimal control vector with equality and inequality state constraints is stated and demonstrated under suitable assumptions. The mathematical formulation of the quaternary adjoint eqs. associated with the quaternary nonlinear parabolic boundary value problem with state constraints is discovered. The Fréchet derivative of the cost function and the constraints functions are derived. The necessary and sufficient theorems (conditions) for optimality of the quaternary continuous classical boundary optimal control vector problem are stated and demonstrated under suitable assumptions.

Improved lower bound for the complexity of unique shortest vector problem

Unique shortest vector problem (uSVP) plays an important role in lattice based cryptography. Many cryptographic schemes based their security on it. For the cofidence of those applications, it is essential to clarify the complexity of uSVP with different parameters. However, proving the NP-hardness of uSVP appears quite hard. To the state of the art, we are even not able to prove the NP-hardness of uSVP with constant parameters. In this work, we gave a lower bound for the hardness of uSVP with constant parameters, i.e. we proved that uSVP is at least as hard as gap shortest vector problem (GapSVP) with gap of O(sqrt{n/log (n)}), which is in NP cap coAM. Unlike previous works, our reduction works for paramters in a bigger range, especially when the constant hidden by the big-O in GapSVP is smaller than 1.Graphical abstract

A Novel Compute-and-Forward Relaying Method for Multi-Antenna Wireless Relay Networks.

Compute-and-Forward (CoF) is an innovative physical layer network coding strategy, designed to enable receivers in wireless communications to effectively utilize interference. The key idea of CoF is to implement integer combinations based on the codewords from multiple transmitters, rather than decoding individual source codewords. Although CoF is widely used in wireless relay networks, there are still some problems to be solved, such as rank failure, single antenna reception, and the shortest vector problem. In this paper, we introduce a successive extended CoF (SECoF) as a pioneering solution tailored for multi-source, multi-relay, and multi-antenna wireless relay networks. First, we analyze the traditional CoF, and design a SECoF method combining the concepts of matrix projection and successive interference cancellation, which overcomes the problem of CoF rate tending to zero and rank failure and improves the network performance. Secondly, we obtain an approximate solution to the integer-value coefficient vectors by using the LLL lattice-based resolution algorithm. In addition, we deduce the corresponding concise formulas of SECoF. Simulation results show that the SECoF has strong robustness and the approaches outperform the state-of-the-art methods in terms of computation rate, rank failure probability, and outage probability.

Computing Generalized Convolutions Faster Than Brute Force

In this paper, we consider a general notion of convolution. Let D be a finite domain and let D^n be the set of n-length vectors (tuples) of D. Let f :Dtimes Drightarrow D be a function and let oplus _f be a coordinate-wise application of f. The f-Convolution of two functions g,h :D^n rightarrow {-M,ldots ,M} is (g⊛fh)(v):=∑vg,vh∈Dns.t.v=vg⊕fvhg(vg)·h(vh)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} (g \\mathbin {\\circledast _{f}}h)(\ extbf{v}) {:}{=}\\sum _{\\begin{array}{c} \ extbf{v}_g,\ extbf{v}_h \\in D^n\\\\ \ ext {s.t. } \ extbf{v}= \ extbf{v}_g \\oplus _f \ extbf{v}_h \\end{array}} g(\ extbf{v}_g) \\cdot h(\ extbf{v}_h) \\end{aligned}$$\\end{document}for every textbf{v}in D^n. This problem generalizes many fundamental convolutions such as Subset Convolution, XOR Product, Covering Product or Packing Product, etc. For arbitrary function f and domain D we can compute f-Convolution via brute-force enumeration in widetilde{{mathcal {O}}}(|D|^{2n} cdot textrm{polylog}(M)) time. Our main result is an improvement over this naive algorithm. We show that f-Convolution can be computed exactly in widetilde{{mathcal {O}}}( (c cdot |D|^2)^{n} cdot textrm{polylog}(M)) for constant c {:}{=}3/4 when D has even cardinality. Our main observation is that a cyclic partition of a function f :Dtimes Drightarrow D can be used to speed up the computation of f-Convolution, and we show that an appropriate cyclic partition exists for every f. Furthermore, we demonstrate that a single entry of the f-Convolution can be computed more efficiently. In this variant, we are given two functions g,h :D^n rightarrow {-M,ldots ,M} alongside with a vector textbf{v}in D^n and the task of the f-Query problem is to compute integer (g mathbin {circledast _{f}}h)(textbf{v}). This is a generalization of the well-known Orthogonal Vectors problem. We show that f-Query can be computed in widetilde{{mathcal {O}}}(|D|^{frac{omega }{2} n} cdot textrm{polylog}(M)) time, where omega in [2,2.372) is the exponent of currently fastest matrix multiplication algorithm.

Quaternary Boundary Optimal Control Problem Dominating by Quaternary Nonlinear Parabolic System

In this paper, our purpose is to study the quaternary continuous classical boundary optimal control vector problem (QCCBOCVP) dominated by the quaternary nonlinear parabolic boundary value problem (QNLPBVP). Under suitable assumptions and with given quaternary continuous classical boundary control vector (QCCBCV), the existence theorem for a unique quaternary state vector solution (QSVS) of the weak form (WF) for the QNLPBVP is stated and demonstrated via the Method of Galerkin and the first compactness theorem. Furthermore, the continuity of the Lipchitz operator between the QSVS of the WF for the QLPBVP and the corresponding QCCBCV is proved. The existence of a quaternary continuous classical boundary optimal control vector (QCCBOVC) is stated and demonstrated under suitable assumptions.

A vectorial problem with thin free boundary

We consider the vectorial analogue of the thin free boundary problem introduced by Caffarelli et al. (J Eur math Soc 12:1151-1179, 2010) as a realization of a nonlocal version of the classical Bernoulli problem. We study optimal regularity, nondegeneracy, and density properties of local minimizers. Via a blow-up analysis based on a Weiss type monotonicity formula, we show that the free boundary is the union of a “regular” and a “singular” part. Finally we use a viscosity approach to prove C^{1,alpha } regularity of the regular part of the free boundary.

Robust Space-Time Adaptive Processing Method for GNSS Receivers in Coherent Signal Environments

In the coherent signal environments caused by multipath propagation, the interference suppression performance of the global navigation satellite systems (GNSS) receivers decreases sharply. In this paper, a robust space-time adaptive processing (STAP) method for GNSS receivers is proposed to suppress interferences in coherent signal environments, by using the modified space-time two-dimensional iterative adaptive approach (ST2D-IAA) spectrum estimation. This method applies the IAA algorithm to the ST2D signal model of GNSS receivers, and further modifies the ST2D-IAA algorithm to accurately estimate the power spectrum and noise power simultaneously. The space-time interference-plus-noise covariance matrix (STINCM) is reconstructed by using the estimated power spectrum and noise power in the interference angle region. Based on the reconstructed STINCM, we construct the STAP beamforming optimization problem for the space-time steering vector (STSV) error vector, and further correct the STSV of GNSS signal. Finally, the weight vector of STAP beamforming is calculated by using the reconstructed STINCM and the corrected STSV of GNSS signal. Simulation results show that the proposed method can suppress interferences in coherent signal environments and outperforms the current methods.

Inverse vector problem of diffraction by inhomogeneous body with a piecewise smooth permittivity

Inverse vector problem of diffraction by inhomogeneous body with a piecewise smooth permittivity

Parameter optimization of SVM algorithm for predicting physical parameters of SF6–Cu mixture plasma under local thermodynamic equilibrium

The physical parameters of SF6–Cu mixture plasma are necessary for arc calculation simulation. The calculation of these parameters is very difficult, but the prediction of the corresponding parameters using the existing database is one of the methods used to solve this difficult problem. The support vector regression (SVR) algorithm can effectively deal with the high-dimensional input vector problem and is widely used in the prediction field. To address the problem that the RBF kernel parameters gamma and penalty coefficient C are difficult to be obtained using the SVR algorithm, which leads to some data not satisfying the prediction accuracy, this paper first uses particle swarm optimization and the gray wolf optimizer to optimize the parameters of SVR, then uses the optimized hyperparameters to predict the data, and finally compares and analyzes the predicted data before and after the optimization. The results show that the optimized SVR parameters obtained using the optimization-seeking algorithm can fit the data better, which verifies the feasibility of the optimization of SVR hyperparameters by the optimization-seeking algorithm.

Lifts for Voronoi Cells of Lattices

Many polytopes arising in polyhedral combinatorics are linear projections of higher-dimensional polytopes with significantly fewer facets. Such lifts may yield compressed representations of polytopes, which are typically used to construct small-size linear programs. Motivated by algorithmic implications for the closest vector problem, we study lifts of Voronoi cells of lattices. We construct an explicit d-dimensional lattice such that every lift of the respective Voronoi cell has 2^{Omega (d/{log d})} facets. On the positive side, we show that Voronoi cells of d-dimensional root lattices and their dual lattices have lifts with {{mathcal {O}}}(d) and {{mathcal {O}}}(d log d) facets, respectively. We obtain similar results for spectrahedral lifts.

Multiscale model reduction of finite-difference frequency-domain wave modelling in acoustic media

SUMMARY Frequency-domain wave modelling can easily describe the visco-acoustic behaviour of wave propagation using frequency-dependent velocities. Conventional finite-difference (FD) modelling in the frequency domain is computationally prohibitive for solving the acoustic Helmholtz equation in complicated and large geological models. To reduce the computational cost of traditional FD Helmholtz solvers, we develop a multiscale FD frequency-domain method that uses multiscale basis functions to significantly reduce the dimension of system matrices associated with the Helmholtz equation. Due to the insufficient accuracy of the first-order multiscale basis functions in the case of strongly heterogeneous models, we introduce the multinode coarse-element scheme into the scalar Helmholtz equation, a scheme previously developed in the extended multiscale finite-element method for vector problems. This multinode scheme enables multiscale basis functions to capture accurate fine-scale medium property variations. We use one homogeneous model and two heterogeneous models to validate our multiscale method for accuracy and computational cost. Numerical results demonstrate that our new approach can significantly reduce the time and memory costs of acoustic wave modelling while maintaining accuracy, indicating the great potential of our multiscale method in large-scale modelling applications.

Raman spectral pattern recognition of breast cancer: A machine learning strategy based on feature fusion and adaptive hyperparameter optimization

Raman spectroscopy, as a kind of molecular vibration spectroscopy, provides abundant information for measuring components and molecular structure in the early detection and diagnosis of breast cancer. Currently, portable Raman spectrometers have simplified and made equipment application more affordable, albeit at the cost of sacrificing the signal-to-noise ratio (SNR). Consequently, this necessitates a higher recognition rate from pattern recognition algorithms. Our study employs a feature fusion strategy to reduce the dimensionality of high-dimensional Raman spectra and enhance the discriminative information between normal tissues and tumors. In the conducted random experiment, the classifier achieved a performance of over 96% for all three average metrics: accuracy, sensitivity, and specificity. Additionally, we propose a multi-parameter serial encoding evolutionary algorithm (MSEA) and integrate it into the Adaptive Local Hyperplane K-nearest Neighbor classification algorithm (ALHK) for adaptive hyperparameter optimization. The implementation of serial encoding tackles the predicament of parallel optimization in multi-hyperparameter vector problems. To bolster the convergence of the optimization algorithm towards a global optimal solution, an exponential viability function is devised for nonlinear processing. Moreover, an improved elitist strategy is employed for individual selection, effectively eliminating the influence of probability factors on the robustness of the optimization algorithm. This study further optimizes the hyperparameter space through sensitivity analysis of hyperparameters and cross-validation experiments, leading to superior performance compared to the ALHK algorithm with manual hyperparameter configuration.

Regularization of the Vector Problem with Quadratic Criteria of Pareto Optimization

Regularization of the Vector Problem with Quadratic Criteria of Pareto Optimization

On the local everywhere Hölder continuity for weak solutions of a class of not convex vectorial problems of the Calculus of Variations

On the local everywhere Hölder continuity for weak solutions of a class of not convex vectorial problems of the Calculus of Variations