Transform Domain Analysis of Sequences

In cryptanalysis, security of ciphers vis-a-vis attacks is gauged against three criteria of complexities, i.e., computations, memory and time. Some features may not be so apparent in a particular domain, and their analysis in a transformed domain often reveals interesting patterns. Moreover, the complexity criteria in different domains are different and performance improvements are often achieved by transforming the problem in an alternate domain. Owing to the results of coding theory and signal processing, Discrete Fourier Transform (DFT) based attacks have proven to be efficient than algebraic attacks in terms of their computational complexity. Motivated by DFT based attacks, we present a transform domain analysis of Linear Feedback Shift Register(LFSR) based sequence generators. The time and frequency domain behavior of non-linear filter and combiner generators is discussed along with some novel observations based on the Chinese Remainder Theorem (CRT). CRT is exploited to establish patterns in LFSR sequences and underlying cyclic structures of finite fields. Application of DFT spectra attacks on combiner generators is also demonstrated. Our proposed method saves on the last stage computations of selective DFT attacks for combiner generators. The proposed approach is demonstrated on some examples of combiner generators and is scalable to general configuration of combiner generators.Comment: This is a comprehensive report with over 20 page

Intelligent Reflecting Surface with Discrete Phase Shifts: Channel Estimation and Passive Beamforming

In this paper, we consider an intelligent reflecting surface (IRS)-aided single-user system where an IRS with discrete phase shifts is deployed to assist the uplink communication. A practical transmission protocol is proposed to execute channel estimation and passive beamforming successively. To minimize the mean square error (MSE) of channel estimation, we first formulate an optimization problem for designing the IRS reflection pattern in the training phase under the constraints of unit-modulus, discrete phase, and full rank. This problem, however, is NP-hard and thus difficult to solve in general. As such, we propose a low-complexity yet efficient method to solve it sub-optimally, by constructing a near-orthogonal reflection pattern based on either discrete Fourier transform (DFT)-matrix quantization or Hadamard-matrix truncation. Based on the estimated channel, we then formulate an optimization problem to maximize the achievable rate by designing the discrete-phase passive beamforming at the IRS with the training overhead and channel estimation error taken into account. To reduce the computational complexity of exhaustive search, we further propose a low-complexity successive refinement algorithm with a properly-designed initialization to obtain a high-quality suboptimal solution. Numerical results are presented to show the significant rate improvement of our proposed IRS training reflection pattern and passive beamforming designs as compared to other benchmark schemes.Comment: Submitted to IEEE conferenc

Efficient function approximation on general bounded domains using wavelets on a cartesian grid

Fourier extension is an approximation method that alleviates the periodicity requirements of Fourier series and avoids the Gibbs phenomenon when approximating functions. We describe a similar extension approach using regular wavelet bases on a hypercube to approximate functions on subsets of that cube. These subsets may have a general shape. This construction is inherently associated with redundancy which leads to severe ill-conditioning, but recent theory shows that nevertheless high accuracy and numerical stability can be achieved using regularization and oversampling. Regularized least squares solvers, such as the truncated singular value decomposition, that are suited to solve the resulting ill-conditioned and skinny linear system generally have cubic computational cost. We compare several algorithms that improve on this complexity. The improvements benefit from the sparsity in and the structure of the discrete wavelet transform. We present a method that requires $\mathcal O(N)$ operations in 1-D and $\mathcal O(N^{3(d-1)/d})$ in $d$-D, $d>1$. We experimentally show that direct sparse QR solvers appear to be more time-efficient, but yield larger expansion coefficients

Fast Continuous Haar and Fourier Transforms of Rectilinear Polygons from VLSI Layouts

We develop the pruned continuous Haar transform and the fast continuous Fourier series, two fast and efficient algorithms for rectilinear polygons. Rectilinear polygons are used in VLSI processes to describe design and mask layouts of integrated circuits. The Fourier representation is at the heart of many of these processes and the Haar transform is expected to play a major role in techniques envisioned to speed up VLSI design. To ensure correct printing of the constantly shrinking transistors and simultaneously handle their increasingly large number, ever more computationally intensive techniques are needed. Therefore, efficient algorithms for the Haar and Fourier transforms are vital. We derive the complexity of both algorithms and compare it to that of discrete transforms traditionally used in VLSI. We find a significant reduction in complexity when the number of vertices of the polygons is small, as is the case in VLSI layouts. This analysis is completed by an implementation and a benchmark of the continuous algorithms and their discrete counterpart. We show that on tested VLSI layouts the pruned continuous Haar transform is 5 to 25 times faster, while the fast continuous Fourier series is 1.5 to 3 times faster.Comment: 10 pages, 10 figure

Compression, Restoration, Re-sampling, Compressive Sensing: Fast Transforms in Digital Imaging

Transform image processing methods are methods that work in domains of image transforms, such as Discrete Fourier, Discrete Cosine, Wavelet and alike. They are the basic tool in image compression, in image restoration, in image re-sampling and geometrical transformations and can be traced back to early 1970-ths. The paper presents a review of these methods with emphasis on their comparison and relationships, from the very first steps of transform image compression methods to adaptive and local adaptive transform domain filters for image restoration, to methods of precise image re-sampling and image reconstruction from sparse samples and up to "compressive sensing" approach that has gained popularity in last few years. The review has a tutorial character and purpose.Comment: 41 pages, 16 figure

Discrete Gyrator Transforms: Computational Algorithms and Applications

As an extension of the 2D fractional Fourier transform (FRFT) and a special case of the 2D linear canonical transform (LCT), the gyrator transform was introduced to produce rotations in twisted space/spatial-frequency planes. It is a useful tool in optics, signal processing and image processing. In this paper, we develop discrete gyrator transforms (DGTs) based on the 2D LCT. Taking the advantage of the additivity property of the 2D LCT, we propose three kinds of DGTs, each of which is a cascade of low-complexity operators. These DGTs have different constraints, characteristics, and properties, and are realized by different computational algorithms. Besides, we propose a kind of DGT based on the eigenfunctions of the gyrator transform. This DGT is an orthonormal transform, and thus its comprehensive properties, especially the additivity property, make it more useful in many applications. We also develop an efficient computational algorithm to significantly reduce the complexity of this DGT. At the end, a brief review of some important applications of the DGTs is presented, including mode conversion, sampling and reconstruction, watermarking, and image encryption.Comment: Accepted by IEEE Transactions on Signal Processin

Digital Shearlet Transform

Over the past years, various representation systems which sparsely approximate functions governed by anisotropic features such as edges in images have been proposed. We exemplarily mention the systems of contourlets, curvelets, and shearlets. Alongside the theoretical development of these systems, algorithmic realizations of the associated transforms were provided. However, one of the most common shortcomings of these frameworks is the lack of providing a unified treatment of the continuum and digital world, i.e., allowing a digital theory to be a natural digitization of the continuum theory. In fact, shearlet systems are the only systems so far which satisfy this property, yet still deliver optimally sparse approximations of cartoon-like images. In this chapter, we provide an introduction to digital shearlet theory with a particular focus on a unified treatment of the continuum and digital realm. In our survey we will present the implementations of two shearlet transforms, one based on band-limited shearlets and the other based on compactly supported shearlets. We will moreover discuss various quantitative measures, which allow an objective comparison with other directional transforms and an objective tuning of parameters. The codes for both presented transforms as well as the framework for quantifying performance are provided in the Matlab toolbox ShearLab.Comment: arXiv admin note: substantial text overlap with arXiv:1106.205

Two-dimensional nonseparable discrete linear canonical transform based on CM-CC-CM-CC decomposition

As a generalization of the two-dimensional Fourier transform (2D FT) and 2D fractional Fourier transform, the 2D nonseparable linear canonical transform (2D NsLCT) is useful in optics, signal and image processing. To reduce the digital implementation complexity of the 2D NsLCT, some previous works decomposed the 2D NsLCT into several low-complexity operations, including 2D FT, 2D chirp multiplication (2D CM) and 2D affine transformations. However, 2D affine transformations will introduce interpolation error. In this paper, we propose a new decomposition called CM-CC-CM-CC decomposition, which decomposes the 2D NsLCT into two 2D CMs and two 2D chirp convolutions (2D CCs). No 2D affine transforms are involved. Simulation results show that the proposed methods have higher accuracy, lower computational complexity and smaller error in the additivity property compared with the previous works. Plus, the proposed methods have perfect reversibility property that one can reconstruct the input signal/image losslessly from the output.Comment: Accepted by Journal of the Optical Society of America A (JOSA A

Fast evaluation of real and complex exponential sums

Recently, the butterfly approximation scheme and hierarchical approximations have been proposed for the efficient computation of integral transforms with oscillatory and with asymptotically smooth kernels. Combining both approaches, we propose a certain fast Fourier-Laplace transform, which in particular allows for the fast evaluation of polynomials at nodes in the complex unit disk. All theoretical results are illustrated by numerical experiments

Fast and Efficient Sparse 2D Discrete Fourier Transform using Sparse-Graph Codes

We present a novel algorithm, named the 2D-FFAST, to compute a sparse 2D-Discrete Fourier Transform (2D-DFT) featuring both low sample complexity and low computational complexity. The proposed algorithm is based on mixed concepts from signal processing (sub-sampling and aliasing), coding theory (sparse-graph codes) and number theory (Chinese-remainder-theorem) and generalizes the 1D-FFAST 2 algorithm recently proposed by Pawar and Ramchandran [1] to the 2D setting. Concretely, our proposed 2D-FFAST algorithm computes a k-sparse 2D-DFT, with a uniformly random support, of size N = Nx x Ny using O(k) noiseless spatial-domain measurements in O(k log k) computational time. Our results are attractive when the sparsity is sub-linear with respect to the signal dimension, that is, when k -> infinity and k/N -> 0. For the case when the spatial-domain measurements are corrupted by additive noise, our 2D-FFAST framework extends to a noise-robust version in sub-linear time of O(k log4 N ) using O(k log3 N ) measurements. Simulation results, on synthetic images as well as real-world magnetic resonance images, are provided in Section VII and demonstrate the empirical performance of the proposed 2D-FFAST algorithm