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Linear complexity is an important pseudo-random measure of the key stream sequence in a stream cipher system. The 1-error linear complexity is used to measure the stability of the linear complexity, which means the minimal linear complexity of the new sequence by changing one bit of the original key stream sequence. This paper contributes to calculating the exact values of the linear complexity and 1-error linear complexity of the binary key stream sequence with two prime periods defined by Ding–Helleseth generalized cyclotomy. We provide a novel method to solve such problems by employing the discrete Fourier transform and the M–S polynomial of the sequence. Our results show that, by choosing appropriate parameters p and q, the linear complexity and 1-error linear complexity can be no less than half period, which shows that the linear complexity of this sequence not only meets the requirements of cryptography but also has good stability. [ABSTRACT FROM AUTHOR]

Vision transformers (ViTs) have pushed the state-of-the-art for visual perception tasks. The self-attention mechanism underpinning the strength of ViTs has a quadratic complexity in both computation and memory usage. This motivates the development of approximating the self-attention at linear complexity. However, an in-depth analysis in this work reveals that existing methods are either theoretically flawed or empirically ineffective for visual recognition. We identify that their limitations are rooted in the inheritance of softmax based self-attention during approximations, that is, normalizing the scaled dot-product between token feature vectors using the softmax function. As preserving the softmax operation challenges any subsequent linearization efforts. By this insight, a family of SOftmax-Free Transformers (SOFT) are proposed. Specifically, a Gaussian kernel function is adopted to replace the dot-product similarity, enabling a full self-attention matrix to be approximated under low-rank matrix decomposition. For computational robustness, we estimate the Moore–Penrose inverse using an iterative Newton–Raphson method in the forward process only, while calculating its theoretical gradients only once in the backward process. To further expand applicability (e.g., dense prediction tasks), an efficient symmetric normalization technique is introduced. Extensive experiments on ImageNet, COCO and ADE20K show that our SOFT significantly improves the computational efficiency of existing ViT variants. With linear complexity, much longer token sequences are permitted by SOFT, resulting in superior trade-off between accuracy and complexity. Code and models are available at https://github.com/fudan-zvg/SOFT. [ABSTRACT FROM AUTHOR]

Since the introduction of the Kolmogorov complexity of binary sequences in the 1960s, there have been significant advancements on the topic of complexity measures for randomness assessment, which are of fundamental importance in theoretical computer science and of practical interest in cryptography. This survey reviews notable research from the past four decades on the linear, quadratic and maximum-order complexities of pseudo-random sequences, and their relations with Lempel–Ziv complexity, expansion complexity, 2-adic complexity and correlation measures. [ABSTRACT FROM AUTHOR]

The linear complexity of new generalized cyclotomic sequences with odd period was estimated. The sequences were defined using generalized cyclotomic classes composite modulo. Conditions sufficient for the existence of binary and non-binary sequences with high linear complexity were obtained. The earlier results on the linear complexity of sequences with the period equal to the power of a prime were generalized.

Linear complexity is an important pseudo-random measure of the keystream sequence in a stream cipher system. The error linear complexity is employed to measure the stability of the linear complexity, which means the minimal linear complexity by changing few bit of the original keystream sequence. This paper contributes to calculating the linear complexity of a famous generalized cyclotomic sequences of period pq defined by Ding and Helleseth. We give the linear complexity of these sequences in a new way, and calculate further the 1-error linear complexity of them. The result shows that the sequences have good stability in terms of linear complexity.

Recently, the transformer-based model e.g., the vision transformer (ViT) has been extensively used in computer vision tasks. The superior performance of the ViT leads to the requirement of an enormous dataset and the complexity of calculating self-attention between patches is quadratic in nature. To acknowledge these two concerns, this paper proposes a novel shifted patch tokenization swin transformer (SPT-Swin) for the image classification task. The shifted patch tokenization (SPT) compensates for the data deficiency by increasing the data samples based on spatial information of the image patches while the swin transformer provides linear computational complexity by calculating self-attention between the shifted window based patches. For model validation, the SPT-Swin framework is trained on popular benchmark image datasets such as ImageNet-1K, CIFAR-10 and CIFAR-100, and the classification accuracies are found 89.45%, 95.67% and 92.95% respectively. Moreover, the comparative analysis of the proposed model with the existing state-of-the-art models shows that the classification performances are improved by 7.05%, 4.14%, and 8.30% for the ImageNet-1K, CIFAR-10 and CIFAR-100 datasets respectively. Therefore, our proposed SPT-based data augmentation technique with the core swin transformer model could be a data-efficient linear complex-able model for future computer vision tasks.

Boolean functions are crucial primitive in block cipher and are also used to design pseudorandom sequences. They play a crucial role in the design of symmetric cryptography and its analysis, and the study on the cryptographic properties of Boolean functions is a hotspot in cryptography. The hidden weighted bit functions (HWBF) are paid attention since they have many “good” cryptographic measures. However, there are no results on their linear complexity in the literature. Therefore, this paper discusses a family of binary sequences of period [2n]built by using [n-]variable HWBF (hidden weighted bit functions). It is proven that such sequences are balanced with maximal linear complexity using mathematical method. The 2-error linear complexity of the sequences is also determined in terms of the Hasse derivative and Lucas congruence. When [n(mod4)∈{0,1,3}], the values of the 2-error linear complexity are maximal. Results indicate that such sequences are of “good” cryptographic measures.

The detection of community structures is a very crucial research area. The problem of community detection has received considerable attention from a large portion of the scientific community. More importantly, these articles are spread across a large number of different disciplines, from computer science, to statistics, and social sciences. The analysis of modern social networks becomes rather cumbersome, as their size and number keeps growing larger and larger. Moreover, in the modern communities, users participate in large number of groups. From the network perspective, efficient methods should be developed to automatically identify overlapping communities, that is, communities with overlapping nodes. In this work, we use a probabilistic network model to characterize and identify linked communities with common nodes. The innovative idea in this work is that the communities are represented as Markovian networks with continuously changing states. Each state represents the number of users within a cluster, that have specific characteristic classes. Based on the current state, we introduce a fast, linear on the number of newly added users, approach to estimate the probability of each cluster to be homogeneous in terms of sets of user characteristics and to determine how well the new user fit within a community. Because of the linear computations involved, our proposed probabilistic model can detect communities and overlaps with low execution time and high accuracy, as shown in our experimental results. The experimental results have shown that our probabilistic scheme executes faster and provides more robust communities compared to competitive schemes.

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Considering the spectral properties of images, we propose a new self-attention mechanism with highly reduced computational complexity, up to a linear rate. To better preserve edges while promoting similarity within objects, we propose individualized processes over different frequency bands. In particular, we study a case where the process is merely over low-frequency components. By ablation study, we show that low frequency self-attention can achieve very close or better performance relative to full frequency even without retraining the network. Accordingly, we design and embed novel plug-and-play modules to the head of a CNN network that we refer to as FsaNet. The frequency self-attention 1) requires only a few low frequency coefficients as input, 2) can be mathematically equivalent to spatial domain self-attention with linear structures, 3) simplifies token mapping ($1\times 1$ convolution) stage and token mixing stage simultaneously. We show that frequency self-attention requires 87.29% ~ 90.04% less memory, 96.13% ~ 98.07% less FLOPs, and 97.56% ~ 98.18% in run time than the regular self-attention. Compared to other ResNet101-based self-attention networks, FsaNet achieves a new state-of-the-art result (83.0% mIoU) on Cityscape test dataset and competitive results on ADE20k and VOCaug. FsaNet can also enhance MASK R-CNN for instance segmentation on COCO. In addition, utilizing the proposed module, Segformer can be boosted on a series of models with different scales, and Segformer-B5 can be improved even without retraining. Code is accessible at https://github.com/zfy-csu/FsaNet. [ABSTRACT FROM AUTHOR]

Finding a provably correct subquadratic synchronization algorithm for many filesystem replicas is one of the main theoretical problems in operational transformation (OT) and conflict-free replicated data types (CRDT) frameworks. Based on the algebraic theory of filesystems, which incorporates non-commutative filesystem commands natively, we developed and built a proof-of-concept implementation of an algorithm suite which synchronizes an arbitrary number of replicas. The result is provably correct, and the synchronized system is created in linear space and time after an initial sorting phase. It works by identifying conflicting command pairs and requesting one of the commands to be removed. The method can be guided to reach any of the theoretically possible synchronized states. The algorithm also allows asynchronous usage. After the client sends a synchronization request, the local replica remains available for further modifications. When the synchronization instructions arrive, they can be merged with the changes made since the synchronization request. The suite also works on filesystems with a directed acyclic graph-based path structure in place of the traditional tree-like arrangement. Consequently, our algorithms apply to filesystems with hard or soft links as long as the links create no loops. [ABSTRACT FROM AUTHOR]

A construction of binary sequences with period 4N and optimal autocorrelation magnitude has been investigated based on sampling and interleaving technique. We determine the exact value of the linear complexity of the constructed sequences according to the deep relationship among the characteristic polynomials, and show it is 2N+2. Moreover, we determine the 2-adic complexity of these sequences by the autocorrelation function, and show it can attain the maximum value. Results show that such sequences can resist both the Berlekamp-Massey attack and the Rational Approximation Algorithm, in addition are good for communication.

Jing et al. dealed with all possible Whiteman generalized cyclotomic binary sequences $ s(a, b, c) $ with period $ N = pq $, where $ (a, b, c) \in \{0, 1\}^3 $ and $ p, q $ are distinct odd primes (Jing et al. arXiv:2105.10947v1, 2021). They have determined the autocorrelation distribution and the 2-adic complexity of these sequences in a unified way by using group ring language and a version of quadratic Gauss sums. In this paper, we determine the linear complexity and the 1-error linear complexity of $ s(a, b, c) $ in details by using the discrete Fourier transform (DFT). The results indicate that the linear complexity of $ s(a, b, c) $ is large enough and stable in most cases.

Pseudorandom sequences with large linear complexity have been widely applied in electronic countermeasures, mobile communication and cryptography. Linear complexity is considered as a primary security criterion to measure the unpredictability of pseudorandom sequences. This paper presents the linear complexity and minimal polynomial of a new family of binary sequences derived from polynomial quotients modulo an odd prime $p$ in general case. The results indicate that the sequences have high linear complexity, which means they can resist the linear attack against pseudo-noise or stream ciphers. Moreover, we generalize the result to the polynomial quotients modulo a power of $p$ in general case. Finally, we design a Gpqs stream cipher generator based on the generalized binary pseudorandom sequences to implement the sequences in hardware.

In this paper, we propose the specific recursion formula for the generalized labeled multi-Bernoulli filter based on the track-before-detect strategy (GLMB-TBD) using a belief propagation algorithm. The proposed method aims to track multiple weak targets with superior performance. Compared to the Murty algorithm-based and Gibbs sampling-based implementation of GLMB-TBD filter, the proposed algorithm improves the tracking accuracy of multiple weak targets without pruning operation to preserve the relevant association information. The superior performance in tracking accuracy of the algorithm is validated for simulated scenarios using OSPA(2) metric. More importantly, the simulation results demonstrate that the proposed algorithm outputs both the Gibbs sampling-based version and Murty algorithm-based version in computational cost due to linear complex in the number of both Bernoulli components and measurements. [ABSTRACT FROM AUTHOR]

Sidelnikov sequences, a kind of cyclotomic sequences with many desired properties such as low correlation and variable alphabet sizes, can be employed to construct a polyphase sequence family that has many applications in high-speed data communications. Recently, cyclotomic numbers have been used to investigate the linear complexity of Sidelnikov sequences, mainly about binary ones, although the limitation on the orders of the available cyclotomic numbers makes it difficult. This paper continues to study the linear complexity over $\mathbb {F}_{M}$ of $M$ -ary Sidelnikov sequence of period $q-1$ using Hasse derivative, which implies $q=p^{m}$ , $m\geq 1$ and $M|(q-1)$. The $t$ th Hasse derivative formulas are presented in terms of cyclotomic numbers, and some upper bounds on the linear complexity for $M=2$ and 3 are obtained only with some additional restrictions on $q$. Furthermore, concrete illustrations for several families of these sequences, such as $q\equiv 1\pmod {2}$ and $q\equiv 1\pmod {3}$ , show these upper bounds are tight and reachable; especially for $q=2\times 3^{\lambda }+1 (1\leq \lambda \leq 20)$ , the exact linear complexities over $\mathbb {F}_{3}$ of the ternary Sidelnikov sequences are determined; and it turns out that all the linear complexities of the sequences considered are very close to their periods. [ABSTRACT FROM AUTHOR]

A class of binary sequences with period 2p is constructed using generalized cyclotomic classes, and their linear complexity, minimal polynomial over F q as well as 2-adic complexity are determined using Gauss period and group ring theory. The results show that the linear complexity of these sequences attains the maximum when p ≡ ± 1 (mod 8) and is equal to p+1 when p ≡ ± 3 (mod 8) over extension field. Moreover, the 2-adic complexity of these sequences is maximum. According to Berlekamp–Massey(B–M) algorithm and the rational approximation algorithm (RAA), these sequences have quite good cryptographic properties in the aspect of linear complexity and 2-adic complexity. [ABSTRACT FROM AUTHOR]

Binary PN-sequences generated by LFSRs exhibit good statistical properties; however, due to their intrinsic linearity, they are not suitable for cryptographic applications. In order to break such a linearity, several approaches can be implemented. For example, one can interleave several PN-sequences to increase the linear complexity. In this work, we present a deep randomness study of the resultant sequences of interleaving binary PN-sequences coming from different characteristic polynomials with the same degree. We analyze the period and the linear complexity, as well as many other important cryptographic properties of such sequences. [ABSTRACT FROM AUTHOR]

Finding a provably correct subquadratic synchronization algorithm for many filesystem replicas is one of the main theoretical problems in operational transformation (OT) and conflict-free replicated data types (CRDT) frameworks. Based on the algebraic theory of filesystems, which incorporates non-commutative filesystem commands natively, we developed and built a proof-of-concept implementation of an algorithm suite which synchronizes an arbitrary number of replicas. The result is provably correct, and the synchronized system is created in linear space and time after an initial sorting phase. It works by identifying conflicting command pairs and requesting one of the commands to be removed. The method can be guided to reach any of the theoretically possible synchronized states. The algorithm also allows asynchronous usage. After the client sends a synchronization request, the local replica remains available for further modifications. When the synchronization instructions arrive, they can be merged with the changes made since the synchronization request. The suite also works on filesystems with a directed acyclic graph-based path structure in place of the traditional tree-like arrangement. Consequently, our algorithms apply to filesystems with hard or soft links as long as the links create no loops.

Let 1 < g1 < . . . < gφ(p−1) < p−1 be the ordered primitive roots modulo p. We study the pseudorandomness of the binary sequence (sn) defined by sn ≡ gn+1 + gn+2mod 2, n = 0,1, . . . In particular, we study the balance, linear complexity and 2-adic complexity of (sn). We show that for a typical p the sequence (sn) is quite unbalanced. However, there are still infinitely many p such that (sn) is very balanced. We also prove similar results for the distribution of longer patterns. Moreover, we give general lower bounds on the linear complexity and 2-adic complexity of (sn) and state sufficient conditions for attaining their maximums. Hence, for carefully chosen p, these sequences are attractive candidates for cryptographic applications. [ABSTRACT FROM AUTHOR]

We solve the problem of generating a uniform Poisson-disk sampling that is both maximal and unbiased over bounded non-convex domains. To our knowledge this is the first provably correct algorithm with time and space dependent only on the number of points produced. Our method has two phases, both based on classical dart-throwing. The first phase uses a background grid of square cells to rapidly create an unbiased, near-maximal covering of the domain. The second phase completes the maximal covering by calculating the connected components of the remaining uncovered voids, and by using their geometry to efficiently place unbiased samples that cover them. The second phase converges quickly, overcoming a common difficulty in dart-throwing methods. The deterministic memory is O(n) and the expected running time is O(n log n), where n is the output size, the number of points in the final sample. Our serial implementation verifies that the log n dependence is minor, and nearly O(n) performance for both time and memory is achieved in practice. We also present a parallel implementation on GPUs to demonstrate the parallel-friendly nature of our method, which achieves 2.4x the performance of our serial version.

Binary sequences are algebraic structures currently used as security elements in Internet of Things devices, sensor networks, e-commerce, and cryptography. In this work, a contribution to the evaluation of such sequences is introduced. In fact, we present a novel algorithm to compute a fundamental parameter for this kind of structure: the linear complexity, which is related to the predictability (or non-predictability) of the binary sequences. Our algorithm reduced the computation of the linear complexity to just the addition modulo two (XOR logic operation) of distinct terms of the sequence. The performance of this procedure was better than that of other algorithms found in the literature. In addition, the amount of required sequence to perform this computation was more realistic than in the rest of the algorithms analysed. Tables, figures, and numerical results complete the work.

Nowadays, the necessity of electronic information increases rapidly. As a consequence, often, that information needs to be shared among mutually distrustful parties. In this area, private set intersection (PSI) and its variants play an important role when the participants wish to do secret operations on their input sets. Unlike the most modern public key cryptosystems relying on number theoretic problems, lattice‐based cryptographic constructions provide security in the presence of a quantum computer. Consequently, developing PSI and its variants using lattice based cryptosystem becomes an interesting direction for research. This study presents the first size‐hiding post quantum PSI cardinality (PSI‐CA) protocol whose complexity is linear in the size of the sets of the participants. The authors use space‐efficient probabilistic data structure (Bloom filter) as its building block. Further, they extend the authors' PSI‐CA to its authorised version, i.e. authorised PSI‐CA. Security for both of them is achieved in the standard model based on the hardness of the decisional learning with errors problem. [ABSTRACT FROM AUTHOR]

The word-based LFSRs called σ -LFSRs are very attractive as they take advantage of the modern word-based processor and thus increase the throughput. Secondly, the bitstream produced by σ -LFSR has excellent statistical properties with a high period except for low linear complexity. In order to increase the linear complexity, the concept of both bit-oriented shrinking and self-shrinking generators is introduced in case of σ -LFSRs. In both the cases, the lower bound for the period as well as for the linear complexity of the bitstream are shown to be exponential. Further, we have experimented and investigated more results on the periodicity and statistical properties of the bitstream in self-shrinking σ -LFSRs. This helps to find and prove the exact period of the bitstream produced by self-shrinking generators. [ABSTRACT FROM AUTHOR]

We first introduce a family of binary $pq^{2}$ -periodic sequences based on the Euler quotients modulo $pq$ , where $p$ and $q$ are two distinct odd primes and $p$ divides $q-1$. The minimal polynomials and linear complexities are determined for the proposed sequences provided that $2^{q-1} \not \equiv 1 \mod {q^{2}}$. The results show that the proposed sequences have high linear complexities. [ABSTRACT FROM AUTHOR]

It is known that Hall’s sextic residue sequence has some desirable features of pseudorandomness: an ideal two-level autocorrelation and linear complexity of the order of magnitude of its period $p$. Here we study its correlation measure of order $k$ and show that it is, up to a constant depending on $k$ and some logarithmic factor, of order of magnitude $p^{1/2}$ , which is close to the expected value for a random sequence of length $p$. Moreover, we derive from this bound a lower bound on the $N$ th maximum order complexity of order of magnitude $\log p$ , which is the expected order of magnitude for a random sequence of length $p$. [ABSTRACT FROM AUTHOR]

Linear complexity is an important property to measure the unpredictability of pseudo-random sequences. Trace representation is helpful for analyzing cryptography properties of pseudo-random sequences. In this paper, a class of new Ding generalized cyclotomic binary sequences of order two with period pq is constructed based on the new segmentation of Ding Helleseth generalized cyclotomy. Firstly, the linear complexity and minimal polynomial of the sequences are investigated. Then, their trace representation is given. It is proved that the sequences have larger linear complexity and can resist the attack of the Berlekamp–Massey algorithm. This paper also confirms that generalized cyclotomic sequences with good randomness may be obtained by modifying the characteristic set of generalized cyclotomy.

Based on the Ding-generalized cyclotomy,a new class of generalized cyclotomic sequences with length pm over the finite field of power of odd prime order was constructed,and the sequence was balanced.The linear complexity of the sequences was determined using the relationship between h and p and the theory of polynomial over finite field.It is shown that the sequence has good linear complexity,and it can resist attacks from the application of the Berlekamp-Massey algorithm.

H2-matrix constitutes a general mathematical framework for efficient computation of both partial-differential-equation (PDE) and integral-equation (IE)-based operators. Existing linear-complexity H2 matrix-matrix product (MMP) algorithm lacks explicit accuracy control, while controlling accuracy without compromising linear complexity is challenging. In this article, we develop an accuracy controlled H2 MMP algorithm by instantaneously changing the cluster bases during the matrix product computation based on prescribed accuracy. Meanwhile, we retain the computational complexity of the overall algorithm to be linear. Different from the existing H2 MMP algorithm where formatted multiplications are performed using the original cluster bases, in the proposed algorithm, all additions and multiplications are either exact or computed based on prescribed accuracy. Furthermore, the original H2-matrix structure is preserved in the matrix product. While achieving optimal complexity for constant-rank matrices, the computational complexity of the proposed algorithm is also minimized for variable-rank H2-matrices. For example, it has a complexity of O(NlogN) for computing electrically large volume IEs, where N is matrix size. The proposed work serves as a fundamental arithmetic in the development of fast solvers for large-scale electromagnetic analysis. Applications to both large-scale capacitance extraction and electromagnetic scattering problems involving millions of unknowns on a single core have demonstrated the accuracy and efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]

This paper presents an efficient quadratic programming (QP) decoder via the alternating direction method of multipliers (ADMM) technique, called QP-ADMM, for binary low-density parity-check (LDPC) codes. Its main contents are as follows: first, we relax the maximum likelihood (ML) decoding problem to a non-convex quadratic program. Then, we develop an ADMM solving algorithm for the formulated non-convex QP decoding model. In the proposed QP-ADMM decoder, complex Euclidean projections onto the check polytope are eliminated and variables in each updated step can be solved analytically in parallel. Moreover, it is proved that the proposed ADMM algorithm converges to a stationary point of the non-convex QP problem under the assumption of sequence convergence. We also verify that the proposed decoder satisfies the favorable property of the all-zeros assumption. Furthermore, by exploiting the inside structures of the QP model, the complexity of the proposed algorithm in each iteration is shown to be linear in terms of LDPC code length. Simulation results demonstrate the effectiveness of the proposed QP-ADMM decoder. [ABSTRACT FROM AUTHOR]

In this paper, we construct two generalized cyclotomic binary sequences of period 2 p m based on the generalized cyclotomy and compute their linear complexity, showing that they are of high linear complexity when m ≥ 2 . [ABSTRACT FROM AUTHOR]

A fast nested cross approximation (NCA) algorithm is developed in this paper for solving large-scale electromagnetic problems. Different from the existing NCA, the proposed method does not rely on the projection of the basis functions onto the dummy interpolation points to select pivots of each cluster. Instead, a purely algebraic and kernel-independent algorithm is developed to find the row and column pivots of all clusters in $\mathcal {O}(N\log {}N)$ complexity for constant-rank cases with controlled accuracy. This algorithm is then further extended to an $\mathcal {O}(N)$ NCA algorithm, which includes a bottom-up tree traversal for finding the local pivots of each cluster, followed by a top-down procedure to take into account the far field of each cluster. The proposed method has a reduced complexity compared to that reported in the mathematical literature. The resultant nested representation constitutes an $\mathcal {H}^{2}$ -matrix representation of the original dense system of equations, whose solution can be obtained in linear complexity in both iterative and direct solvers. The method is also applicable to variable rank cases, but the complexity therein depends on the rank’s relationship with $N$. Various numerical experiments have demonstrated the accuracy and computational performance of the proposed algorithms. [ABSTRACT FROM AUTHOR]

Given an infinite word x = x0x1x2 · · · ∈ AN over some finite alphabet A, the factor complexity px(n) counts the number of distinct factors of x of each given length n, i.e., the number of distinct blocks xixi+1 · · · xi+n−1 ∈ An occurring in x. The factor complexity provides a useful measure of the extent of randomness of x: periodic words have bounded factor complexity while digit expansions of normal numbers have maximal complexity. In this paper we obtain a new characterization of infinite words x of sublinear complexity, namely we show that px(n) = O(n) if and only if there exists a set S ⊆ A∗ of bounded complexity (meaning lim sup pS(n) < +∞) such that each factor w of x is a concatenation of two elements of S, i.e., w = uv with u, v ∈ S. In the process we introduce the notions of marker words and marker sets which are both new and may be of independent interest. Marker sets defined by right special factors constitute the key tool needed to split each factor of an infinite word of linear complexity into two pieces. [ABSTRACT FROM AUTHOR]

Abstract The Linear Complexity Test is a statistical test for verifying the randomness of a binary sequence produced by a random number generator (RNG). It is the most time-consuming test in the widely used randomness testing suite that was published by the National Institute of Standards and Technology (NIST). The slow performance of the original Linear Complexity Test implementation is one of the major hurdles in the RNG testing process. In this work, we present a parallelized implementation of the Linear Complexity Test for GPU computation. We incorporate two levels of parallelism and various design optimization approaches to accelerate the test execution on modern GPU architectures. To further enhance the performance, we also create a hybrid computation approach that uses both CPU and GPU simultaneously. We achieve a speedup of more than 4000 times over the original Linear Complexity Test implementation from NIST (27 times over the previous best implementation of the test). Highlights • Parallel implementation of the Linear Complexity Test for GPU computation. • Achieved a speedup of more than several orders of magnitude. • Decision mechanism for finding the ideal workload amount per thread. • A hybrid computing approach that simultaneously utilizes CPU and GPU. [ABSTRACT FROM AUTHOR]

This paper examines the linear complexity of a family of generalized cyclotomic binary sequences of period p n recently proposed by Xiao et al. (Des Codes Cryptogr, 2017, 10.1007/s10623-017-0408-7), where a conjecture about the linear complexity in the special case that f = 2 r for a positive integer r was made. We prove the conjecture and also extend the result to more general even integers f. [ABSTRACT FROM AUTHOR]

In this paper, the method of calculating the k-variance linear complexity distribution with 2n-periodical sequences by the Games-Chan algorithm and sieve approach is affirmed for its generality. The main idea of this method is to decompose a binary sequence into some subsequences of critical requirements, hence the issue to find k-variance linear complexity distribution with 2n-periodical sequences becomes a combinatorial problem of these binary subsequences. As a result, we compute the whole calculating formulas on the k-variance linear complexity with 2n-periodical sequences of linear complexity less than 2n for k = 4, 5. With combination of results in the whole calculating formulas on the 3-variance linear complexity with 2n-periodical binary sequences of linear complexity 2n, we completely solve the problem of the calculating function distributions of 4-variance linear complexity with 2n-periodical sequences elegantly, which significantly improves the results in the relating references. [ABSTRACT FROM AUTHOR]

In this work, we construct a well-posed first-order system least squares (FOSLS) simultaneously space-time formulation of parabolic PDEs. Using an adaptive wavelet solver, this problem is solved with the best possible rate in linear complexity. Thanks to the use of a basis that consists of tensor products of wavelets in space and time, this rate is equal to that when solving the corresponding stationary problem. Our findings are illustrated by numerical results. [ABSTRACT FROM AUTHOR]

Linear complexity is an important criterion to characterize the unpredictability of pseudo-random sequences, and large linear complexity corresponds to high cryptographic strength. Pseudo-random Sequences with a large linear complexity property are of importance in many domains. In this paper, based on the theory of inverse Gray mapping, two classes of new generalized cyclotomic quaternary sequences with period pq are constructed, where pq is a product of two large distinct primes. In addition, we give the linear complexity over the residue class ring Z4 via the Hamming weights of their Fourier spectral sequence. The results show that these two kinds of sequences have large linear complexity.

The output sequence of the shrinking generator can be considered as an interleaving of determined shifted versions of a single PN -sequence. In this paper, we present a study of the interleaving of a PN-sequence and shifted versions of itself. We analyze some important cryptographic properties as the period and the linear complexity in terms of the shifts. Furthermore, we determine the total number of the interleaving sequences that achieve each possible value of the linear complexity.

Spatio-temporal graph neural networks (GNN) that model inter-variable dependencies with learned graph structures for multivariate time series forecasting have received attention due to their superior performance. However, existing methods exhibit quadratic complexity, limiting the GNN's capability to handle a large number of variables. Moreover, the message-passing mechanism employed by spatio-temporal GNNs can be seen as a type of Laplacian smoothing, which weakens the inherent evolution patterns of nodes and overlooks the dynamic impact of a node's own information and that of its neighbors on future outcomes. In this paper, we propose a novel approach called the Dynamic Personalized Graph Neural Network (DPGNN), which presents a graph learning framework with linear complexity. To facilitate linearization, we conceptualize the spatio-temporal GNN as an information flow from the source (raw node features) to the sink (new node features). And the flow conservation principle is employed to generate the information attention mechanism to effectively model the dependency between variables. To address the second issue, inspired by PageRank, we design a dynamic personalized graph convolution scheme that explicitly quantifies the impact of each node on the subsequent results based on the dynamic graph signals at each propagation step. Experimental results on six benchmark datasets showcase the superiority of our approach over state-of-the-art methods, including both pre-defined graph structures and graph learning techniques. [Display omitted] • Spatio-temporal graph networks with linear complexity learn graph automatically. • Dynamic personalized graph convolution enhances the influence of the nodes' own. • Experiments on six datasets demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

DNA fragment assembly represents an important challenge to the development of efficient and practical algorithms due to the large number of elements to be assembled. In this study, we present some graph theoretical linear time algorithms to solve the problem. To achieve linear time complexity, a heap with constant time operations was developed, for the special case where the edge weights are integers and do not depend on the problem size. The experiments presented show that modified classical graph theoretical algorithms can solve the DNA fragment assembly problem efficiently.

The Berlekamp-Massey algorithm finds the shortest linear feedback shift register for a binary input sequence. A wide range of applications like cryptography and digital signal processing use this algorithm. This research proposes novel parallel mechanisms offered by heterogeneous CPU and GPU hardwares in order to achieve the best possible performance for BMA. The proposed bitwise implementation of the BMA algorithm is almost 35 times faster than state of the art implementations. This further improvement is achieved by using SIMD instructions which provides data level parallelism. This new approach can be 4.6 and 35 times faster than a bitwise CPU and state of the art implementations, respectively. In order to achieve the highest possible speedup over a multi-core structure, a multi-threading implementation is introduced in this research. By leveraging on OpenMP we were able to obtain a speedup of 10 times over 12 cores server. The GPU device with thousands of processing cores can bring great speedup over the best CPU implementation. Two other parallel mechanisms offered by GPU are concurrent kernel execution and streaming. They achieve 14.5 and 2.2 times of speedup compared to CPU serial and typical CUDA implementations, respectively. Also, the performance of the openMP code with SIMD instructions is compared with GPU stream implementation. The effectiveness of the proposed method is evaluated in a real world error correction application and it achieves 6.8 times of speedup. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we define the linear complexity for multidimensional sequences over finite fields, generalizing the one-dimensional case. We give some lower and upper bounds, valid with large probability, for the linear complexity and k -error linear complexity of multidimensional periodic sequences. [ABSTRACT FROM AUTHOR]

Binary sequences with optimal autocorrelation and large linear complexity have important applications in cryptography and communications. Very recently, a class of binary sequences of period 4p with optimal autocorrelation was proposed by interleaving four suitable Ding-Helleseth-Lam sequences (Des. Codes Cryptogr., 10.1007/s10623-017-0398-5), where p is an odd prime with p≡1(mod4). The objective of this paper is to determine the minimal polynomial and the linear complexity of this class of binary optimal sequences via a sequence polynomial approach. It turns out that this class of sequences has quite good linear complexity. [ABSTRACT FROM AUTHOR]

Let p be an odd prime and w

New generalized cyclotomic binary sequences of period p2 are proposed in this paper, where p is an odd prime. The sequences are almost balanced and their linear complexity is determined. The result shows that the proposed sequences have very large linear complexity if p is a non-Wieferich prime. [ABSTRACT FROM AUTHOR]