5,040 research outputs found

    Linear Complexity and Autocorrelation of two Classes of New Interleaved Sequences of Period 2N2N

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    The autocorrelation and the linear complexity of a key stream sequence in a stream cipher are important cryptographic properties. Many sequences with these good properties have interleaved structure, three classes of binary sequences of period 4N4N with optimal autocorrelation values have been constructed by Tang and Gong based on interleaving certain kinds of sequences of period NN. In this paper, we use the interleaving technique to construct a binary sequence with the optimal autocorrelation of period 2N2N, then we calculate its autocorrelation values and its distribution, and give a lower bound of linear complexity. Results show that these sequences have low autocorrelation and the linear complexity satisfies the requirements of cryptography

    On the kk-error linear complexity for pnp^n-periodic binary sequences via hypercube theory

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    The linear complexity and the kk-error linear complexity of a binary sequence are important security measures for key stream strength. By studying binary sequences with the minimum Hamming weight, a new tool named as hypercube theory is developed for pnp^n-periodic binary sequences. In fact, hypercube theory is based on a typical sequence decomposition and it is a very important tool in investigating the critical error linear complexity spectrum proposed by Etzion et al. To demonstrate the importance of hypercube theory, we first give a standard hypercube decomposition based on a well-known algorithm for computing linear complexity and show that the linear complexity of the first hypercube in the decomposition is equal to the linear complexity of the original sequence. Second, based on such decomposition, we give a complete characterization for the first decrease of the linear complexity for a pnp^n-periodic binary sequence ss. This significantly improves the current existing results in literature. As to the importance of the hypercube, we finally derive a counting formula for the mm-hypercubes with the same linear complexity.Comment: 16 pages. arXiv admin note: substantial text overlap with arXiv:1309.1829, arXiv:1312.692

    Some New Balanced and Almost Balanced Quaternary Sequences with Low Autocorrelation

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    Quaternary sequences of both even and odd period having low autocorrelation are studied. We construct new families of balanced quaternary sequences of odd period and low autocorrelation using cyclotomic classes of order eight, as well as investigate the linear complexity of some known quaternary sequences of odd period. We discuss a construction given by Chung et al. in "New Quaternary Sequences with Even Period and Three-Valued Autocorrelation" [IEICE Trans. Fundamentals Vol. E93-A, No. 1 (2010)] first by pointing out a slight modification (thereby obtaining new families of balanced and almost balanced quaternary sequences of even period and low autocorrelation), then by showing that, in certain cases, this slight modification greatly simplifies the construction given by Shen et al. in "New Families of Balanced Quaternary Sequences of Even Period with Three-level Optimal Autocorrelation" [IEEE Comm. Letters DOI10.1109/LCOMM.2017.26611750 (2017)]. We investigate the linear complexity of these sequences as well

    Characterization of 2n2^n-periodic binary sequences with fixed 3-error or 4-error linear complexity

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    The linear complexity and the kk-error linear complexity of a sequence have been used as important security measures for key stream sequence strength in linear feedback shift register design. By using the sieve method of combinatorics, the kk-error linear complexity distribution of 2n2^n-periodic binary sequences is investigated based on Games-Chan algorithm. First, for k=2,3k=2,3, the complete counting functions on the kk-error linear complexity of 2n2^n-periodic binary sequences with linear complexity less than 2n2^n are characterized. Second, for k=3,4k=3,4, the complete counting functions on the kk-error linear complexity of 2n2^n-periodic binary sequences with linear complexity 2n2^n are presented. Third, for k=4,5k=4,5, the complete counting functions on the kk-error linear complexity of 2n2^n-periodic binary sequences with linear complexity less than 2n2^n are derived. As a consequence of these results, the counting functions for the number of 2n2^n-periodic binary sequences with the 3-error linear complexity are obtained, and the complete counting functions on the 4-error linear complexity of 2n2^n-periodic binary sequences are obvious.Comment: 7 page

    On the kk-error linear complexity for 2n2^n-periodic binary sequences via Cube Theory

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    The linear complexity and k-error linear complexity of a sequence have been used as important measures of keystream strength, hence designing a sequence with high linear complexity and kk-error linear complexity is a popular research topic in cryptography. In this paper, the concept of stable kk-error linear complexity is proposed to study sequences with stable and large kk-error linear complexity. In order to study k-error linear complexity of binary sequences with period 2n2^n, a new tool called cube theory is developed. By using the cube theory, one can easily construct sequences with the maximum stable kk-error linear complexity. For such purpose, we first prove that a binary sequence with period 2n2^n can be decomposed into some disjoint cubes and further give a general decomposition approach. Second, it is proved that the maximum kk-error linear complexity is 2n−(2l−1)2^n-(2^l-1) over all 2n2^n-periodic binary sequences, where 2l−1≤k<2l2^{l-1}\le k<2^{l}. Thirdly, a characterization is presented about the ttth (t>1t>1) decrease in the kk-error linear complexity for a 2n2^n-periodic binary sequence ss and this is a continuation of Kurosawa et al. recent work for the first decrease of k-error linear complexity. Finally, A counting formula for mm-cubes with the same linear complexity is derived, which is equivalent to the counting formula for kk-error vectors. The counting formula of 2n2^n-periodic binary sequences which can be decomposed into more than one cube is also investigated, which extends an important result by Etzion et al..Comment: 11 pages. arXiv admin note: substantial text overlap with arXiv:1109.4455, arXiv:1108.5793, arXiv:1112.604

    The kk-error linear complexity distribution for 2n2^n-periodic binary sequences

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    The linear complexity and the kk-error linear complexity of a sequence have been used as important security measures for key stream sequence strength in linear feedback shift register design. By studying the linear complexity of binary sequences with period 2n2^n, one could convert the computation of kk-error linear complexity into finding error sequences with minimal Hamming weight. Based on Games-Chan algorithm, the kk-error linear complexity distribution of 2n2^n-periodic binary sequences is investigated in this paper. First, for k=2,3k=2,3, the complete counting functions on the kk-error linear complexity of 2n2^n-periodic balanced binary sequences (with linear complexity less than 2n2^n) are characterized. Second, for k=3,4k=3,4, the complete counting functions on the kk-error linear complexity of 2n2^n-periodic binary sequences with linear complexity 2n2^n are presented. Third, as a consequence of these results, the counting functions for the number of 2n2^n-periodic binary sequences with the kk-error linear complexity for k=2k = 2 and 3 are obtained. Further more, an important result in a recent paper is proved to be not completely correct

    Frequency hopping sequences with optimal partial Hamming correlation

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    Frequency hopping sequences (FHSs) with favorable partial Hamming correlation properties have important applications in many synchronization and multiple-access systems. In this paper, we investigate constructions of FHSs and FHS sets with optimal partial Hamming correlation. We first establish a correspondence between FHS sets with optimal partial Hamming correlation and multiple partition-type balanced nested cyclic difference packings with a special property. By virtue of this correspondence, some FHSs and FHS sets with optimal partial Hamming correlation are constructed from various combinatorial structures such as cyclic difference packings, and cyclic relative difference families. We also describe a direct construction and two recursive constructions for FHS sets with optimal partial Hamming correlation. As a consequence, our constructions yield new FHSs and FHS sets with optimal partial Hamming correlation.Comment: 16 pages. arXiv admin note: text overlap with arXiv:1506.0737

    The 4-error linear complexity distribution for 2n2^n-periodic binary sequences

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    By using the sieve method of combinatorics, we study kk-error linear complexity distribution of 2n2^n-periodic binary sequences based on Games-Chan algorithm. For k=4,5k=4,5, the complete counting functions on the kk-error linear complexity of 2n2^n-periodic balanced binary sequences (with linear complexity less than 2n2^n) are presented. As a consequence of the result, the complete counting functions on the 4-error linear complexity of 2n2^n-periodic binary sequences (with linear complexity 2n2^n or less than 2n2^n) are obvious. Generally, the complete counting functions on the kk-error linear complexity of 2n2^n-periodic binary sequences can be obtained with a similar approach.Comment: 15 pages. arXiv admin note: substantial text overlap with arXiv:1108.5793, arXiv:1112.6047, arXiv:1309.182

    Structure Analysis on the kk-error Linear Complexity for 2n2^n-periodic Binary Sequences

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    In this paper, in order to characterize the critical error linear complexity spectrum (CELCS) for 2n2^n-periodic binary sequences, we first propose a decomposition based on the cube theory. Based on the proposed kk-error cube decomposition, and the famous inclusion-exclusion principle, we obtain the complete characterization of iith descent point (critical point) of the k-error linear complexity for i=2,3i=2,3. Second, by using the sieve method and Games-Chan algorithm, we characterize the second descent point (critical point) distribution of the kk-error linear complexity for 2n2^n-periodic binary sequences. As a consequence, we obtain the complete counting functions on the kk-error linear complexity of 2n2^n-periodic binary sequences as the second descent point for k=3,4k=3,4. This is the first time for the second and the third descent points to be completely characterized. In fact, the proposed constructive approach has the potential to be used for constructing 2n2^n-periodic binary sequences with the given linear complexity and kk-error linear complexity (or CELCS), which is a challenging problem to be deserved for further investigation in future.Comment: 19 pages. arXiv admin note: substantial text overlap with arXiv:1309.1829, arXiv:1310.0132, arXiv:1108.5793, arXiv:1112.604

    Linear Complexity of Geometric Sequences Defined by Cyclotomic Classes and Balanced Binary Sequences Constructed by the Geometric Sequences

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    Pseudorandom number generators are required to generate pseudorandom numbers which have good statistical properties as well as unpredictability in cryptography. An m-sequence is a linear feedback shift register sequence with maximal period over a finite field. M-sequences have good statistical properties, however we must nonlinearize m-sequences for cryptographic purposes. A geometric sequence is a sequence given by applying a nonlinear feedforward function to an m-sequence. Nogami, Tada and Uehara proposed a geometric sequence whose nonlinear feedforward function is given by the Legendre symbol, and showed the period, periodic autocorrelation and linear complexity of the sequence. Furthermore, Nogami et al. proposed a generalization of the sequence, and showed the period and periodic autocorrelation. In this paper, we first investigate linear complexity of the geometric sequences. In the case that the Chan--Games formula which describes linear complexity of geometric sequences does not hold, we show the new formula by considering the sequence of complement numbers, Hasse derivative and cyclotomic classes. Under some conditions, we can ensure that the geometric sequences have a large linear complexity from the results on linear complexity of Sidel'nikov sequences. The geometric sequences have a long period and large linear complexity under some conditions, however they do not have the balance property. In order to construct sequences that have the balance property, we propose interleaved sequences of the geometric sequence and its complement. Furthermore, we show the periodic autocorrelation and linear complexity of the proposed sequences. The proposed sequences have the balance property, and have a large linear complexity if the geometric sequences have a large one.Comment: 20 pages, 3 figures. arXiv admin note: text overlap with arXiv:1709.0516
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