Linear Complexity Of Binary Sequences Research Articles

An Efficient Algorithm to Compute the Linear Complexity of Binary Sequences

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.

On the $ k $-error linear complexity of binary sequences of periods $ p^n $ from new cyclotomy

<abstract><p>In this paper, we study the $ k $-error linear complexity of binary sequences with periods $ p^n $, which are derived from new generalized cyclotomic classes modulo a power of an odd prime $ p $. We establish a recursive relation and then estimate the $ k $-error linear complexity of the binary sequences with periods $ p^n $, the results extend the case $ p^2 $ that has been studied in an earlier work of Wu et al. at 2019. Our results show that the $ k $-error linear complexity of these sequences does not decrease dramatically for $ k < (p^{n}-p^{n-1})/2 $.</p></abstract>

Improvements on k-error linear complexity of q-ary sequences derived from Euler quotients

In this paper, we generalize the results about the k-error linear complexity of binary sequences derived from Euler quotients modulo on the q-ary sequences. We also continue the study of q-ary sequences derived from Euler quotients modulo 2p started by R. Mohammed et al. (LNCS, 11634, 2019) and investigate the k-error linear complexity of sequences derived from Euler quotients with a period 2pr . In particular, we generalize the results of Wu et al. (IEEE Access, 8, 2020) about k-error the binary sequences derived by Euler quotients modulo 2p on the same sequences with a period 2pr .

On k-Error Linear Complexity of Binary Sequences Derived From Euler Quotients Modulo 2p

We consider the $k$-error linear complexity of binary sequences derived from Eluer quotients modulo $2p$ ($p>3$ is an odd prime), recently introduced by J. Zhang and C. Zhao. We adopt certain decimal sequences to determine the values of $k$-error linear complexity for all $k>0$. Our results indicate that such sequences have good stability from the viewpoint of cryptography.

Linear complexity of binary sequences with optimal autocorrelation magnitude of length 4N

In this paper, we study the linear complexity of series of binary sequences with optimal autocorrelation magnitude of length 4N. These sequences are obtained from almost-perfect binary sequences and binary sequences with optimal autocorrelation of length N. The construction of these sequences were presented E.I. Krengel and P.V. Ivanov. We show that considered sequences have the high linear complexity. Also we derive the 1-error linear complexity of these sequences.

The linear complexity of binary sequences of length 2p with optimal three-level autocorrelation

In this paper we derive the linear complexity of binary sequences of length 2p with optimal three-level autocorrelation. These almost balanced and balanced sequences are constructed by cyclotomic classes of order four using a method presented by Ding et al. We investigate the linear complexity of above-mentioned sequences over the finite fields of different orders.

On the k-error linear complexity of binary sequences derived from polynomial quotients

We investigate the $k$-error linear complexity of $p^2$-periodic binary sequences defined from the polynomial quotients (including the well-studied Fermat quotients), which is defined by $$ q_{p,w}(u)\equiv \frac{u^w-u^{wp}}{p} \bmod p ~ \mathrm{with} 0 \le q_{p,w}(u) \le p-1, ~u\ge 0, $$ where $p$ is an odd prime and $1\le w<p$. Indeed, first for all integers $k$, we determine exact values of the $k$-error linear complexity over the finite field $\F_2$ for these binary sequences under the assumption of f2 being a primitive root modulo $p^2$, and then we determine their $k$-error linear complexity over the finite field $\F_p$ for either $0\le k<p$ when $w=1$ or $0\le k<p-1$ when $2\le w<p$. Theoretical results obtained indicate that such sequences possess `good' error linear complexity.

On the Error Linear Complexity Spectrum of Binary Sequences with Period of Power of Two

The properties of error linear complexity of binary sequences with period of power of two are studied in this paper. Using the Games-Chan algorithm as main tool, accurate formulas of the minimum value k for which the k-error linear complexity is strictly less than the first and second critical error linear complexity are provided respectively.

A New Method to Compute the 2-Adic Complexity of Binary Sequences

In this paper, a new method is presented to compute the 2-adic complexity of pseudo-random sequences. With this method, the 2-adic complexities of all the known sequences with ideal 2-level autocorrelation are determined in a unified way. Results show that their 2-adic complexities equal their periods. In other words, their 2-adic complexities attain the maximum. In addition, 2-adic complexities of two classes of optimal autocorrelation sequences with period N ≡ 1mod4, namely Legendre sequences and Ding-Helleseth-Lam sequences, are investigated. This method also can be used to compute the linear complexity of binary sequences regarded as sequences over other finite fields.

Trace representation and linear complexity of binary sequences derived from Fermat quotients

We describe the trace representations of two families of binary sequences derived from Fermat quotients modulo an odd prime $p$ (one is the binary threshold sequences, the other is the Legendre-Fermat quotient sequences) via determining the defining pairs of all binary characteristic sequences of cosets, which coincide with the sets of pre-images modulo $p^2$ of each fixed value of Fermat quotients. From the defining pairs, we can obtain an earlier result of linear complexity for the binary threshold sequences and a new result of linear complexity for the Legendre-Fermat quotient sequences under the assumption of $2^{p-1}\not\equiv 1 \bmod {p^2}$.

Linear complexity of binary sequences with interleaved structure

In this study, the minimal polynomials and the linear complexity of interleaved binary sequences are investigated. Both the linear complexity and the minimal polynomials of low correlation zone sequences constructed by Zhou et al. are completely determined. Besides, an open problem proposed by Li and Tang is discussed. At last, a sufficient condition and a necessary condition are presented about when the linear complexity of the interleaved sequences constructed by Tang et al. attains the maximum.

Distribution of one-error linear complexity of binary sequences for arbitrary prime period

Complexity measures for sequences, such as the linear complexity and the k-error linear complexity, play an important role in stream ciphers. This contribution studies the distribution of 1-error linear complexity of binary sequences with arbitrary prime period. For any odd prime N, the authors present all possible values of 1-error linear complexity of N-periodic binary sequences, and derive the exact formulas to count the number of N-periodic binary sequences with any given 1-error linear complexity.

Linear complexity of binary sequences derived from Euler quotients with prime-power modulus

We extend the definition of binary threshold sequences from Fermat quotients to Euler quotients modulo pr with odd prime p and r⩾1. Under the condition of 2p−1≢1(modp2), we present exact values of the linear complexity by defining cyclotomic classes modulo pn for all 1⩽n⩽r. The linear complexity is very close to the period and is of desired value for cryptographic purpose. We also present a lower bound on the linear complexity for the case of 2p−1≡1(modp2).

On the Linear Complexity of Binary Sequences of Period $4N$ With Optimal Autocorrelation Value/Magnitude

Three classes of binary sequences of period 4N with optimal autocorrelation value/magnitude have been constructed by Tang and Gong based on interleaving certain kinds of sequences of period N , i.e., the Legendre sequence, twin-prime sequence and generalized GMW sequence. In this paper, by means of sequence polynomials of the underlying sequences, the properties of roots of the corresponding sequence polynomials of the interleaved sequences with period 4N and optimal autocorrelation value/magnitude are discussed in the splitting field of xN-1 . As a consequence, both the minimal polynomials and linear complexities of these three classes of sequences are completely determined except for the case of the sequences obtained from the generalized GMW sequences. For the latter, the minimal polynomial and linear complexity can be specially obtained if the sequence is constructed based on m-sequences instead of generalized GMW sequences.

Asymptotic analysis on the normalized k-error linear complexity of binary sequences

Linear complexity and k-error linear complexity are the important measures for sequences in stream ciphers. This paper discusses the asymptotic behavior of the normalized k-error linear complexity $${L_{n,k}(\underline{s})/n}$$ of random binary sequences $${\underline{s}}$$ , which is based on one of Niederreiter's open problems. For k = n ?, where 0 ? ? ? 1/2 is a fixed ratio, the lower and upper bounds on accumulation points of $${L_{n,k}(\underline{s})/n}$$ are derived, which holds with probability 1. On the other hand, for any fixed k it is shown that $${\lim_{n\rightarrow\infty} L_{n,k}(\underline{s})/n = 1/2}$$ holds with probability 1. The asymptotic bounds on the expected value of normalized k-error linear complexity of binary sequences are also presented.

The Linear Complexity of Binary Sequences With Optimal Autocorrelation

Binary sequences with optimal autocorrelation are needed in many applications. Two constructions of binary sequences with optimal autocorrelation of period N ≡ 0 (mod 4) are investigated. The two constructions are powerful and generic in the sense that many classes of binary sequences with optimal autocorrelation could be obtained from binary sequences with ideal autocorrelation. General results on the minimal polynomials of these binary sequences are derived. Based on the results, both the linear complexities and the minimal polynomials are determined.

On Linear Complexity of Binary Sequences Generated Using Matrix Recurrence Relation Defined Over Z4

On Linear Complexity of Binary Sequences Generated Using Matrix Recurrence Relation Defined Over Z4

On Linear Complexity of Binary Sequences Generated Using Matrix Recurrence Relation Defined Over Z4

This paper discusses the linear complexity property of binary sequences generated using matrix recurrence relation defined over Z4. Generally algorithm to generate random number is based on recursion with seed value/values. In this paper a linear recursion sequence of matrices or vectors over Z 4 is generated from which random binary sequence is obtained. It is shown that such sequences have large linear complexity.

On the linear complexity of binary sequences on the basis of biquadratic and sextic residue classes

On the linear complexity of binary sequences on the basis of biquadratic and sextic residue classes

How to determine linear complexity and k-error linear complexity in some classes of linear recurring sequences

Several fast algorithms for the determination of the linear complexity of d-periodic sequences over a finite field \({\mathbb F}_q\), i.e. sequences with characteristic polynomial f(x) = xd − 1, have been proposed in the literature. In this contribution fast algorithms for determining the linear complexity of binary sequences with characteristic polynomial f(x) = (x − 1)d for an arbitrary positive integer d, and \(f(x) = (x^2+x+1)^{2^v}\) are presented. The result is then utilized to establish a fast algorithm for determining the k-error linear complexity of binary sequences with characteristic polynomial \((x^2+x+1)^{2^v}\).