Linear Complexity Of Sequences Research Articles

On the Stability of the Linear Complexity of Some Generalized Cyclotomic Sequences of Order Two

Linear complexity is an important pseudo-random measure of the key stream sequence in a stream cipher system. The 1-error linear complexity is used to measure the stability of the linear complexity, which means the minimal linear complexity of the new sequence by changing one bit of the original key stream sequence. This paper contributes to calculating the exact values of the linear complexity and 1-error linear complexity of the binary key stream sequence with two prime periods defined by Ding–Helleseth generalized cyclotomy. We provide a novel method to solve such problems by employing the discrete Fourier transform and the M–S polynomial of the sequence. Our results show that, by choosing appropriate parameters p and q, the linear complexity and 1-error linear complexity can be no less than half period, which shows that the linear complexity of this sequence not only meets the requirements of cryptography but also has good stability.

Linear Complexity of Sequences on Koblitz Curves of Genus 2

Abstract In this paper, we consider the hyperelliptic analogue of the Frobenius endomorphism generator and show that it produces sequences with large linear complexity on the Jacobian of genus 2 curves.

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>

АНАЛИЗ ЛИНЕЙНОЙ СЛОЖНОСТИ Q-ИЧНЫХ ОБОБЩЕННЫХ ЦИКЛОТОМИЧЕСКИХ ПОСЛЕДОВАТЕЛЬНОСТЕЙ ПЕРИОДА pn

The article presents the analysis of the linear complexity of periodic q-ary sequences when changing k of their terms per period. Sequences are formed on the basis of new generalized cyclotomy modulo equal to the degree of an odd prime. There has been obtained a recurrence relation and an estimate of the change in the linear complexity of these sequences, where q is a primitive root modulo equal to the period of the sequence. It can be inferred from the results that the linear complexity of these sequences does not sign ificantly decrease when k is less than half the period. The study summarizes the results for the binary case obtained earlier.

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 .

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.

Character values of the Sidelnikov-Lempel-Cohn-Eastman sequences

Binary sequences with good autocorrelation properties and large linear complexity are useful in stream cipher cryptography. The Sidelnikov-Lempel-Cohn-Eastman (SLCE) sequences have nearly optimal autocorrelation. However, the problem of determining the linear complexity of the SLCE sequences is still open. It is well known that one can gain insight into the linear complexity of a sequence if one can say something about the divisors of the gcd of a certain pair of polynomials associated with the sequence. Helleseth and Yang (IEEE Trans. Inf. Theory 49(6), 1548–1552 2002), Kyureghyan and Pott (Des. Codes Crypt. 29, 149–164 2003) and Meidl and Winterhof (Des. Codes Crypt. 8, 159–178 2006) were able to obtain some results of this type for the SLCE sequences. Kyureghyan and Pott (Des. Codes Crypt. 29, 149–164 2003) mention that it would be nice to obtain more such results. We derive new divisibility results for the SLCE sequences in this paper. Our approach is to exploit the fact that character values associated with the SLCE sequences can be expressed in terms of a certain type of Jacobi sum. By making use of known evaluations of Gauss and Jacobi sums in the “pure” and “small index” cases, we are able to obtain new insight into the linear complexity of the SLCE sequences.

Expansion complexity and linear complexity of sequences over finite fields

The linear complexity is a measure for the unpredictability of a sequence over a finite field and thus for its suitability in cryptography. In 2012, Diem introduced a new figure of merit for cryptographic sequences called expansion complexity. We study the relationship between linear complexity and expansion complexity. In particular, we show that for purely periodic sequences both figures of merit provide essentially the same quality test for a sufficiently long part of the sequence. However, if we study shorter parts of the period or nonperiodic sequences, then we can show, roughly speaking, that the expansion complexity provides a stronger test. We demonstrate this by analyzing a sequence of binomial coefficients modulo $p$. Finally, we establish a probabilistic result on the behavior of the expansion complexity of random sequences over a finite field.

Linear complexity of quaternary sequences with odd period and low autocorrelation

Equivalence between two classes of quaternary sequences with odd period and best known autocorrelation are proved. A lower bound on the linear complexity of these sequences is presented. It is shown that the quaternary sequences have large linear complexity to resist Reeds and Sloane algorithm attack effectively.

The minimal polynomial of a sequence obtained from the componentwise linear transformation of a linear recurring sequence

Let S=(s1,s2,…,sm,…) be a linear recurring sequence with terms in GF(qn) and T be a linear transformation of GF(qn) over GF(q). Denote T(S)=(T(s1),T(s2),…,T(sm),…). In this paper, we first present counter examples to show that the main result in [A.M. Youssef, G. Gong, On linear complexity of sequences over GF(2n), Theoret. Comput. Sci., 352 (2006) 288–292] is not correct in general since Lemma 3 in that paper is incorrect. Then we determine the minimal polynomial of T(S) if the canonical factorization of the minimal polynomial of S without multiple roots is known and thus present the solution to the main problem which was considered in the above paper but incorrectly solved. Additionally, as a special case, we determine the minimal polynomial of T(S) if the minimal polynomial of S is primitive. Finally, we give an upper bound on the linear complexity of T(S) when T exhausts all possible linear transformations of GF(qn) over GF(q). This bound is tight in some cases.

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

The linear complexity of sequences is an important measure of the cryptographic strength of key streams used in stream ciphers. The instability of linear complexity caused by changing a few symbols of sequences can be measured using k-error linear complexity. In their SETA 2006 paper, Fu et al. (SETA, pp. 88---103, 2006) studied the linear complexity and the 1-error linear complexity of 2 n -periodic binary sequences to characterize such sequences with fixed 1-error linear complexity. In this paper we study the linear complexity and the k-error linear complexity of 2 n -periodic binary sequences in a more general setting using a combination of algebraic, combinatorial, and algorithmic methods. This approach allows us to characterize 2 n -periodic binary sequences with fixed 2- or 3-error linear complexity. Using this characterization we obtain the counting function for the number of 2 n -periodic binary sequences with fixed k-error linear complexity for k = 2 and 3.

Joint linear complexity of multisequences consisting of linear recurring sequences

The linear complexity of sequences is one of the important security measures for stream cipher systems. Recently, in the study of vectorized stream cipher systems, the joint linear complexity of multisequences has been investigated. In this paper, we study the joint linear complexity of multisequences consisting of linear recurring sequences. The expectation and variance of the joint linear complexity of random multisequences consisting of linear recurring sequences are determined. These results extend the corresponding results on the expectation and variance of the joint linear complexity of random periodic multisequences. Then we enumerate the multisequences consisting of linear recurring sequences with fixed joint linear complexity. A general formula for the appropriate counting function is derived. Some convenient closed-form expressions for the counting function are determined in special cases. Furthermore, we derive tight upper and lower bounds on the counting function in general. Some interesting relationships among the counting functions of certain cases are established. The generating polynomial for the distribution of joint linear complexities is determined. The proofs use new methods that enable us to obtain results of great generality.

Linear Complexity of Sequences under Different Interpretations

In this paper we study relationships between the linear complexities of a sequence when treated as a sequence over two distinct fields. We obtain bounds for one linear complexity in the form of a constant multiple of the other, where the constant depends only on the fields, not on the particular sequence.

The expectation and variance of the joint linear complexity of random periodic multisequences

The linear complexity of sequences is one of the important security measures for stream cipher systems. Recently, in the study of vectorized stream cipher systems, the joint linear complexity of multisequences has been investigated. By using the generalized discrete Fourier transform for multisequences, Meidl and Niederreiter determined the expectation of the joint linear complexity of random N-periodic multisequences explicitly. In this paper, we study the expectation and variance of the joint linear complexity of random periodic multisequences. Several new lower bounds on the expectation of the joint linear complexity of random periodic multisequences are given. These new lower bounds improve on the previously known lower bounds on the expectation of the joint linear complexity of random periodic multisequences. By further developing the method of Meidl and Niederreiter, we derive a general formula and a general upper bound for the variance of the joint linear complexity of random N-periodic multisequences. These results generalize the formula and upper bound of Dai and Yang for the variance of the linear complexity of random periodic sequences. Moreover, we determine the variance of the joint linear complexity of random periodic multisequences with certain periods.

On the linear complexity of product sequences of linear recurring sequences

In this note we find the linear complexity of sequences which are the product of two different phases of a single binary linear recurring sequence. Our result extends known facts for linear recurring sequences with primitive polynomials to linear recurring sequences with generator polynomials that are products of primitive polynomials.

Generalized GMW Quadriphase Sequences Satisfying the Welch Bound with Equality

This paper gives new families of quadriphase sequences obtained from large linear complexity sequences over Z4 by making use of generalized permutation monomials over Galois rings. The construction of these sequences can be seen as a generalization of the binary GMW sequence construction and hence they are referred to as GGMW sequences over Z4. The GGMW families satisfy the Welch bound on inner products with equality and it is shown that the root mean square of all the crosscorrelations and out-of-phase autocorrelations (θrms), is approximately equal to the quantity \(\); L being the period of the sequences. However, θmax, the maximum magnitude of periodic crosscorrelation and out-of-phase autocorrelation, deviates from the optimal value of \(\). Computer results suggest that the number of crosscorrelation values which deviate from the optimal value of \(\) is small. The weight structure of these sequences is the same as those of m-sequences over Z4. The linear complexity (LC) of the sequences is computed using a generalized Blahuts theorem on the LC of sequences over Z4.

Estudio de algunas Secuencias pseudoaleatorias de aplicación criptográfica

Pseudorandom binary sequences are required in stream ciphers and other applications of modern communication systems. In the first case it is essential that the sequences be unpredictable. The linear complexity of a sequence is the amount of it required to define the remainder. This work addresses the problem of the analysis and computation of the linear complexity of certain pseudorandom binary sequences. Finally we conclude some characteristics of the nonlinear function that produces the sequences to guarantee a minimum linear complexity.

Root Counting, the DFT and the Linear Complexity of Nonlinear Filtering

The method of root counting is a well established technique in the study of the linear complexity of sequences. Recently, Massey and Serconek [11] have introduced a Discrete Fourier Transform approach to the study of linear complexity. In this paper, we establish the equivalence of these two approaches. The power of the DFT methods are then harnessed to re-derive Rueppel‘s Root Presence Test, a key result in the theory of filtering of m-sequences, in an elegant and concise way. The application of Rueppel‘s Test is then extended to give lower bounds on linear complexity for new classes of filtering functions.

On evaluating the linear complexity of a sequence of least period 2 n

The linear complexity of a periodic binary sequence is the length of the shortest linear feedback shift register that can be used to generate that sequence. When the sequence has least period 2 n ,n≥0, there is a fast algorithm due to Games and Chan that evaluates this linear complexity. In this paper a related algorithm is presented that obtains the linear complexity of the sequence requiring, on average for sequences of period 2 n ,n≥0, no more than 2 parity checks sums.