5,040 research outputs found
Linear Complexity and Autocorrelation of two Classes of New Interleaved Sequences of Period
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 with optimal autocorrelation values have been constructed by
Tang and Gong based on interleaving certain kinds of sequences of period .
In this paper, we use the interleaving technique to construct a binary sequence
with the optimal autocorrelation of period , 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 -error linear complexity for -periodic binary sequences via hypercube theory
The linear complexity and the -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 -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
-periodic binary sequence . This significantly improves the current
existing results in literature. As to the importance of the hypercube, we
finally derive a counting formula for the -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
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 -periodic binary sequences with fixed 3-error or 4-error linear complexity
The linear complexity and the -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 -error linear complexity distribution of -periodic
binary sequences is investigated based on Games-Chan algorithm.
First, for , the complete counting functions on the -error linear
complexity of -periodic binary sequences with linear complexity less than
are characterized. Second, for , the complete counting functions
on the -error linear complexity of -periodic binary sequences with
linear complexity are presented. Third, for , the complete
counting functions on the -error linear complexity of -periodic binary
sequences with linear complexity less than are derived. As a consequence
of these results, the counting functions for the number of -periodic
binary sequences with the 3-error linear complexity are obtained, and the
complete counting functions on the 4-error linear complexity of -periodic
binary sequences are obvious.Comment: 7 page
On the -error linear complexity for -periodic binary sequences via Cube Theory
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 -error linear complexity is a popular
research topic in cryptography. In this paper, the concept of stable -error
linear complexity is proposed to study sequences with stable and large
-error linear complexity. In order to study k-error linear complexity of
binary sequences with period , a new tool called cube theory is developed.
By using the cube theory, one can easily construct sequences with the maximum
stable -error linear complexity. For such purpose, we first prove that a
binary sequence with period can be decomposed into some disjoint cubes
and further give a general decomposition approach. Second, it is proved that
the maximum -error linear complexity is over all
-periodic binary sequences, where . Thirdly, a
characterization is presented about the th () decrease in the -error
linear complexity for a -periodic binary sequence 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 -cubes with the same
linear complexity is derived, which is equivalent to the counting formula for
-error vectors. The counting formula of -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 -error linear complexity distribution for -periodic binary sequences
The linear complexity and the -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 , one could convert the computation of
-error linear complexity into finding error sequences with minimal Hamming
weight. Based on Games-Chan algorithm, the -error linear complexity
distribution of -periodic binary sequences is investigated in this paper.
First, for , the complete counting functions on the -error linear
complexity of -periodic balanced binary sequences (with linear complexity
less than ) are characterized. Second, for , the complete counting
functions on the -error linear complexity of -periodic binary sequences
with linear complexity are presented. Third, as a consequence of these
results, the counting functions for the number of -periodic binary
sequences with the -error linear complexity for 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
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 -periodic binary sequences
By using the sieve method of combinatorics, we study -error linear
complexity distribution of -periodic binary sequences based on Games-Chan
algorithm. For , the complete counting functions on the -error linear
complexity of -periodic balanced binary sequences (with linear complexity
less than ) are presented. As a consequence of the result, the complete
counting functions on the 4-error linear complexity of -periodic binary
sequences (with linear complexity or less than ) are obvious.
Generally, the complete counting functions on the -error linear complexity
of -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 -error Linear Complexity for -periodic Binary Sequences
In this paper, in order to characterize the critical error linear complexity
spectrum (CELCS) for -periodic binary sequences, we first propose a
decomposition based on the cube theory. Based on the proposed -error cube
decomposition, and the famous inclusion-exclusion principle, we obtain the
complete characterization of th descent point (critical point) of the
k-error linear complexity for . Second, by using the sieve method and
Games-Chan algorithm, we characterize the second descent point (critical point)
distribution of the -error linear complexity for -periodic binary
sequences. As a consequence, we obtain the complete counting functions on the
-error linear complexity of -periodic binary sequences as the second
descent point for . 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
-periodic binary sequences with the given linear complexity and -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
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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