In this paper, we construct two generalized cyclotomic binary sequences of period 2p(m) based on the generalized cyclotomy and compute their linear complexity, showing that they are of high linear complexity when m &g...
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In this paper, we construct two generalized cyclotomic binary sequences of period 2p(m) based on the generalized cyclotomy and compute their linear complexity, showing that they are of high linear complexity when m >= 2.
During the last two decades, many kinds of periodic sequences with good pseudorandom properties have been constructed from classical and generalized cyclotomic classes, and used as keystreams for stream ciphers and se...
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During the last two decades, many kinds of periodic sequences with good pseudorandom properties have been constructed from classical and generalized cyclotomic classes, and used as keystreams for stream ciphers and secure communications. Among them are a family DH-GCS(d) of generalized cyclotomic sequences on the basis of Ding and Helleseth's generalized cyclotomy, of length pq and order d=gcd(p-1,q-1) for distinct odd primes p and q. The linear complexity (or linear span), as a valuable measure of unpredictability, is precisely determined for DH-GCS(8) in this paper. Our approach is based on Edemskiy and Antonova's computation method with the help of explicit expressions of Gaussian classical cyclotomic numbers of order 8. Our result for d = 8 is compatible with Yan's low bound (pq - 1)/2 on the linear complexity for any order d, which is high enough to resist attacks of the Berlekamp-Massey algorithm. Finally, we include SageMath codes to illustrate the validity of our result by examples.
${\mathcal{ H}}{2}$ -matrix constitutes a general mathematical framework for efficient computation of both partial-differential-equation (PDE) and integral-equation (IE)-based operators. Existing linear-complexity ${\...
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${\mathcal{ H}}<^>complexity$ -matrix constitutes a general mathematical framework for efficient computation of both partial-differential-equation (PDE) and integral-equation (IE)-based operators. Existing linear-complexity ${\mathcal{ H}}<^>complexity$ 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 ${\mathcal{ H}}<^>complexity$ 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 ${\mathcal{ H}}<^>complexity$ 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 ${\mathcal{ H}}<^>complexity$ -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 ${\mathcal{ H}}<^>complexity$ -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.
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...
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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 Z(4) via the Hamming weights of their Fourier spectral sequence. The results show that these two kinds of sequences have large linear complexity.
Given an infinite word x = x(0)x(1)x(2) .... is an element of A(N) over some finite alphabet A, the factor complexity p(x)(n) counts the number of distinct factors of x of each given length n, i.e., the number of dist...
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Given an infinite word x = x(0)x(1)x(2) .... is an element of A(N) over some finite alphabet A, the factor complexity p(x)(n) counts the number of distinct factors of x of each given length n, i.e., the number of distinct blocks x(i)x(ii+1) ... x(i+n-1) is an element of A(n) 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 p(x)(n) = O(n) if and only if there exists a set S subset of A * of bounded complexity (meaning lim sup p(S)(n) < + infinity) such that each factor w of x is a concatenation of two elements of S, i.e., w = uv with u, v is an element of 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.
This paper examines the linear complexity of a family of generalized cyclotomic binary sequences of period pn recently proposed by Xiao et al. (Des Codes Cryptogr, 2017, 10.1007/s10623-017-0408-7), where a conjecture ...
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This paper examines the linear complexity of a family of generalized cyclotomic binary sequences of period pn 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=2r for a positive integer r was made. We prove the conjecture and also extend the result to more general even integers f.
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 ...
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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). (C) 2019 Elsevier Inc. All rights reserved.
Recently, a conjecture on the linear complexity of a new class of generalized cyclotomic binary sequences of period pr was proposed by Xiao et al. (Des Codes Cryptogr 86(7):1483-1497, 2018). Later, for the case f bein...
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Recently, a conjecture on the linear complexity of a new class of generalized cyclotomic binary sequences of period pr was proposed by Xiao et al. (Des Codes Cryptogr 86(7):1483-1497, 2018). Later, for the case f being the form 2a with a1, Vladimir Edemskiy proved the conjecture (arXiv:1712.03947). In this paper, under the assumption of 2p-1?1,f)=1, the conjecture proposed by Xiao et al. is proved for a general f by using the Euler quotient. Actually, a generic construction of pr-periodic binary sequences based on the generalized cyclotomy is introduced in this paper, which admits a flexible support set and contains Xiao's construction as a special case, and then an efficient method to compute the linear complexity of the sequence by the generic construction is presented, based on which the conjecture proposed by Xiao et al. could be easily proved under the aforementioned assumption.
In this paper, new classes of binary generalized cyclotomic sequences of period 2p(m+1)q(n+1) are constructed. These sequences are balanced. We calculate the linear complexity of the constructed sequences with a simpl...
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In this paper, new classes of binary generalized cyclotomic sequences of period 2p(m+1)q(n+1) are constructed. These sequences are balanced. We calculate the linear complexity of the constructed sequences with a simple method. The results show that the linear complexity of such sequences attains the maximum.
In the paper of Kyureghyan and Pott (Des Codes Cryptogr 29:149-164, 2003), the linear feedback polynomials of the Sidelnikov-Lempel-Cohn-Eastman sequences were determined for some special cases. We found that Corollar...
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In the paper of Kyureghyan and Pott (Des Codes Cryptogr 29:149-164, 2003), the linear feedback polynomials of the Sidelnikov-Lempel-Cohn-Eastman sequences were determined for some special cases. We found that Corollary 4 and Theorem 2 of that paper are wrong because there exist counterexamples for these two results. In this note, we give some counterexamples of Corollary 4 and Theorem 2, and correct them by readopting the negation of the condition of Lemma 5 as their necessary and sufficient conditions.
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