137 research outputs found
A fast algorithm for determining the linear complexity of periodic sequences
A fast algorithm is presented for determining the linear complexity and the
minimal polynomial of periodic sequences over GF(q) with period q n p m, where
p is a prime, q is a prime and a primitive root modulo p2. The algorithm
presented here generalizes both the algorithm in [4] where the period of a
sequence over GF(q) is p m and the algorithm in [5] where the period of a
binary sequence is 2 n p m . When m=0, the algorithm simplifies the generalized
Games-Chan algorithm.Comment: 7 page
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
An algorithm for the k-error linear complexity of a sequence with period 2pn over GF(q)
The union cost is used, so that an efficient algorithm for computing the
k-error linear complexity of a sequence with period 2pn over GF(q) is
presented, where p and q are odd primes, and q is a primitive root of modulo
p2.Comment: 6 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
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
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
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
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
Periodic sequences with stable -error linear complexity
The linear complexity of a sequence has been used as an important measure of
keystream strength, hence designing a sequence which possesses high linear
complexity and -error linear complexity is a hot topic in cryptography and
communication. Niederreiter first noticed many periodic sequences with high
-error linear complexity over GF(q). In this paper, the concept of stable
-error linear complexity is presented to study sequences with high -error
linear complexity. By studying linear complexity of binary sequences with
period , the method using cube theory to construct sequences with maximum
stable -error linear complexity is presented. It is proved that a binary
sequence with period can be decomposed into some disjoint cubes. The cube
theory is a new tool to study -error linear complexity. Finally, it is
proved that the maximum -error linear complexity is over all
-periodic binary sequences, where
Embedding Constructions of Tail-Biting Trellises for Linear Block Codes
In this paper, embedding construction of tail-biting trellises for linear
block codes is presented. With the new approach of constructing tail-biting
trellises, most of the study of tail-biting trellises can be converted into the
study of conventional trellises. It is proved that any minimal tail-biting
trellis can be constructed by the recursive process of embedding constructions
from the well-known Bahl-Cocke-Jelinek-Raviv (BCJR) constructed conventional
trellises. Furthermore, several properties of embedding constructions of
tail-biting trellises are discussed. Finally, we give four sufficient
conditions to reduce the maximum state-complexity of a trellis with one peak
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