Large Linear Complexity Research Articles

New Binary Sequences Derived from Euler Quotients Modulo pq and Their Generalizations

Pseudorandom sequences with large linear complexity can resist the linear attack. The trace representation plays an important role in analysis and design of pseudorandom sequences. In this letter, we present the construction of a family of new binary sequences derived from Euler quotients modulo pq, where pq is a product of two primes and p divides q-1. Firstly, the linear complexity of the sequences are investigated. It is proved that the sequences have larger linear complexity and can resist the attack of Berlekamp-Massey algorithm. Then, we give the trace representation of the proposed sequences by determining the corresponding defining pair. Moreover, we generalize the result to the Euler quotients modulo pmqn with m≤n. Results indicate that the generalized sequences still have high linear complexity. We also give the trace representation of the generalized sequences by determining the corresponding defining pair. The result will be helpful for the implementation and the pseudorandom properties analysis of the sequences.

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.

Pseudorandom sequences derived from automatic sequences

Many automatic sequences, such as the Thue-Morse sequence or the Rudin-Shapiro sequence, have some desirable features of pseudorandomness such as a large linear complexity and a small well-distribution measure. However, they also have some undesirable properties in view of certain applications. For example, the majority of possible binary patterns never appears in automatic sequences and their correlation measure of order 2 is extremely large. Certain subsequences, such as automatic sequences along squares, may keep the good properties of the original sequence but avoid the bad ones. In this survey we investigate properties of pseudorandomness and non-randomness of automatic sequences and their subsequences and present results on their behaviour under several measures of pseudorandomness including linear complexity, correlation measure of order k, expansion complexity and normality. We also mention some analogs for finite fields.

Linear Complexity of New Binary Sequence Derived From Polynomial Quotients Modulo p in General Case and Their Generalizations

Pseudorandom sequences with large linear complexity have been widely applied in electronic countermeasures, mobile communication and cryptography. Linear complexity is considered as a primary security criterion to measure the unpredictability of pseudorandom sequences. This paper presents the linear complexity and minimal polynomial of a new family of binary sequences derived from polynomial quotients modulo an odd prime <i>p</i> in general case. Results indicate that the sequences have high linear complexity, which means they can resist the linear attack against pseudo-noise or stream ciphers. Moreover, we generalize the result to the polynomial quotients modulo a power of <i>p</i> in general case. Finally, we design a Gpqs stream cipher generator based on the generalized binary pseudorandom sequences.

Linear complexity of some sequences derived from hyperelliptic curves of genus 2

For a given hyperelliptic curve C over a finite field with Jacobian JC, we consider the hyperelliptic analogue of the congruential generator defined by Wn = Wn− 1 + D for n ≥ 1 and D, W0 ∈ JC. We show that curves of genus 2 produce sequences with large linear complexity.

New Generalized Cyclotomic Quaternary Sequences with Large Linear Complexity and a Product of Two Primes Period

Linear complexity is an important criterion to characterize the unpredictability of pseudo-random sequences, and large linear complexity corresponds to high cryptographic strength. Pseudo-random Sequences with a large linear complexity property are of importance in many domains. In this paper, based on the theory of inverse Gray mapping, two classes of new generalized cyclotomic quaternary sequences with period pq are constructed, where pq is a product of two large distinct primes. In addition, we give the linear complexity over the residue class ring Z4 via the Hamming weights of their Fourier spectral sequence. The results show that these two kinds of sequences have large linear complexity.

Generation of Binary Sequences of Length 10230 Bits Having Better Odd and Even Correlation with Large Linear Complexity for Use in Global Navigation Satellites Systems (GNSS) Applications

Generation of Binary Sequences of Length 10230 Bits Having Better Odd and Even Correlation with Large Linear Complexity for Use in Global Navigation Satellites Systems (GNSS) Applications

Linear Complexity of d ‐Ary Sequence Derived from Euler Quotients over GF( q )

For an odd prime p and positive integers r , d such that 0 < d ≤ p r , a generic construction of d -ary sequence based on Euler quotients is presented in this paper. Compared with the known construction, in which the support set of the sequence is fixed and d is usually required to be a prime, the support set of the proposed sequence is flexible and d could be any positive integer less then pr in our construction. Furthermore, the linear complexity of the proposed sequence over prime field GF( q ) with the assumption of q p -1 ≢ 1 mod p 2 is determined. An algorithm of computing the linear complexity of the sequence is also given. Our results indicate that, with some constrains on the support set, the new sequences possess large linear complexities.

On error linear complexity of new generalized cyclotomic binary sequences of period p2

We consider the k-error linear complexity of a new generalized cyclotomic binary sequence of period p2 for an odd prime p. The new sequences were introduced recently by Z. Xiao, X. Zeng, C. Li and T. Helleseth by defining a new kind of generalized cyclotomic classes modulo p2. They proved that the sequences had large linear complexity. In this work, we determine the values of the k-error linear complexity in terms of the theory of Fermat quotients. The results indicate that such sequences have good stability, that is, the linear complexity does not significantly decrease by changing a few terms.

Linear complexity and autocorrelation of a new class of binary generalized cyclotomic sequences of order two and length pqr

Cyclotomic sequences have good cryptographic properties and are closely related to difference sets. This paper proposes a new class of binary generalized cyclotomic sequences of order two and length pqr. Its linear complexity, minimal polynomial, and autocorrelation are investigated. The results show that these sequences have a large linear complexity when 2 e D 1 , which means they can resist the Berlekamp-Massey attack. Furthermore, the autocorrelation values are close to 0 with a probability of approximately 1 — 1/r. Therefore, when r is a big prime, the new sequence has a good autocorrelation.

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.

Linear Complexity of Quaternary Sequences over Z&lt;sub&gt;4&lt;/sub&gt; Based on Ding-Helleseth Generalized Cyclotomic Classes

A family of quaternary sequences over Z4 is defined based on the Ding-Helleseth generalized cyclotomic classes modulo pq for two distinct odd primes p and q. The linear complexity is determined by computing the defining polynomial of the sequences, which is in fact connected with the discrete Fourier transform of the sequences. The results show that the sequences possess large linear complexity and are good sequences from the viewpoint of cryptography.

The linear complexity of a class of binary sequences with optimal autocorrelation

Binary sequences with optimal autocorrelation and large linear complexity have important applications in cryptography and communications. Very recently, a class of binary sequences of period 4p with optimal autocorrelation was proposed by interleaving four suitable Ding–Helleseth–Lam sequences (Des. Codes Cryptogr., https://doi.org/10.1007/s10623-017-0398-5 ), where p is an odd prime with $$p \equiv 1(\bmod 4)$$ . The objective of this paper is to determine the minimal polynomial and the linear complexity of this class of binary optimal sequences via a sequence polynomial approach. It turns out that this class of sequences has quite good linear complexity.

Linear Complexity of the Balanced Polynomial Quotients Sequences

Balanced binary sequences of large linear complexity have series applications in communication systems. In the past, although the sequences derived from polynomial quotients have large linear complexity, but they are not balanced. In this paper, we will construct new sequences which are not only with large linear complexity but also balanced. Meanwhile, this linear complexity reaches the known k-error linear complexity mentioned in [7], which means that the k-error linear complexity as a lower bound is tight.

On the Nth linear complexity of automatic sequences

The Nth linear complexity of a sequence is a measure of predictability. Any unpredictable sequence must have large Nth linear complexity. However, in this paper we show that for q-automatic sequences over Fq the converse is not true.We prove that any (not ultimately periodic) q-automatic sequence over Fq has Nth linear complexity of order of magnitude N. For some famous sequences including the Thue–Morse and Rudin–Shapiro sequence we determine the exact values of their Nth linear complexities. These are non-trivial examples of predictable sequences with Nth linear complexity of largest possible order of magnitude.

Characteristic digit-sum sequences

We introduce a new type of sequences using the sum of coefficients of characteristic polynomials for elements (in particular, primitive elements) in a finite field. These sequences are nonlinear filtering sequences of the well-known m-sequences. We show that they have large linear complexity and large period. We also provide some examples of such binary sequences with good autocorrelation values.

New generalized cyclotomic binary sequences of period $$p^2$$ p 2

New generalized cyclotomic binary sequences of period $p^2$ are proposed in this paper, where $p$ is an odd prime. The sequences are almost balanced and their linear complexity is determined. The result shows that the proposed sequences have very large linear complexity if $p$ is a non-Wieferich prime.

Linear Complexity of Least Significant Bit of Polynomial Quotients

Binary sequences with large linear complexity have been found many applications in communication systems. We determine the linear complexity of a family of p2-periodic binary sequences derived from polynomial quotients modulo an odd prime p. Results show that these sequences have high linear complexity, which means they can resist the linear attack method.

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.

Two constructions of binary sequences with optimal autocorrelation magnitude

In this Letter, two constructions of new binary sequences with optimal autocorrelation magnitude of length 4N derived from binary sequences with optimal autocorrelation of length N = 2 (mod 4) and almost-perfect binary sequences of length 2N using N × 2 interleaved structure is presented. The first construction is to use binary Sidelnikov sequences of length N = pn −1 whereas the second one is to use binary Ding–Helleseth–Martinsen sequences of length N = 2p. The obtained sequences have large linear complexity and can be used in communication and cryptography.