Family Of Binary Sequences Research Articles

Elliptic curve analogues of a pseudorandom generator

Using the discrete logarithm in [7] and [9] a large family of pseudorandom binary sequences was constructed. Here we extend this construction. An interesting feature of this extension is that in certain special cases we get sequences involving points on elliptic curves.

Large families of pseudorandom binary sequences constructed by using the Legendre symbol

Large families of pseudorandom binary sequences constructed by using the Legendre symbol

Measures of pseudorandomness of families of binary lattices, I (Definitions, a construction using quadratic characters)

In the last 15 years a new constructive theory of pseudorandomness of binary sequences has been developed. Later this theory was extended to n dimensions, i.e., to the study of pseudorandomness of binary lattices. In the applications it is not enough to consider single binary sequences, one also needs information on the structure of large families of binary sequences with strong pseudorandom properties. Thus the related notions of family complexity, collision and avalanche effect have been introduced. In this paper our goal is to extend these definitions to binary lattices, and we will present constructions of large families of binary lattices with strong pseudorandom properties such that these families also possess a nice structure.

GMW 수열과 No 수열에 의해서 생성된 이진 수열 분석

In this paper, a family of binary sequences generated by GMW sequences and No sequences is introduced and analyzed. Each sequence within a family has period      ,    and there are   sequences within that family. We obtain auto and cross-correlation values and linear span of the synthesized sequence.

GOWERS UNIFORMITY NORM AND PSEUDORANDOM MEASURES OF THE PSEUDORANDOM BINARY SEQUENCES

Recently there has been much progress in the study of arithmetic progressions. An important tool in these developments is the Gowers uniformity norm. In this paper we study the Gowers norm for pseudorandom binary sequences, and establish some connections between these two subjects. Some examples are given to show that the "good" pseudorandom sequences have small Gowers norm. Furthermore, we introduce two large families of pseudorandom binary sequences constructed by the multiplicative inverse and additive character, and study the pseudorandom measures and the Gowers norm of these sequences by using the estimates of exponential sums and properties of the Vandermonde determinant. Our constructions are superior to the previous ones from some points of view.

On the Correlation Distributions of the Optimal Quaternary Sequence Family ${\cal U}$ and the Optimal Binary Sequence Family ${\cal V}$

Recently, new optimal Families <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\cal S}$</tex></formula> and <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\cal U}$</tex></formula> of quaternary sequences have been presented, and the optimal binary sequence Family <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\cal V}$</tex></formula> obtained from Family <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">${\cal S}$</tex></formula> under Gray map has been investigated as well. The two sequence Families <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\cal U}$</tex></formula> and <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\cal V}$</tex></formula> are optimal with respect to the well-known Sidelnikov bound and Welch bound, but their exact correlation distributions are not known until now. In this paper, their exact correlation distributions are completely determined in some cases by making use of exponential sums and the theory of <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">${\bf Z}_4$</tex></formula> -valued quadratic forms.

Modifications of Modified Jacobi Sequences

The known families of binary sequences having asymptotic merit factor 6.0 are modifications to the families of Legendre sequences and Jacobi sequences. In this paper, we show that at N = pq, there are many suitable modifications other than the Jacobi or modified Jacobi sequences. Furthermore, we will give three new modifications to the character sequences of length N = pq. Based on these new modifications, for any pair of large p and q, we can construct a binary sequence of length 2pq so that such families of sequences have asymptotic merit factor 6.0 without cyclic shifting of the base sequences.

On the generalized large set of Kasami sequences

In this paper, we present a new family of binary Kasami sequences of length 2n − 1 for even integer n = 2m. It generalizes the recently proposed large set $${\mathcal{F}^{(l)}}$$of Kasami sequences by relaxing the restriction on l to a more general case. The presented sequence family takes five nontrival correlation values −1, −1 ± 2m , −1 ± 2m + gcd(m,l), and has family size 23m + 2m or 23m + 2m −1. We also completely determine the distribution of these correlation values.

Construction of pseudorandom binary sequences using additive characters over GF(2 k ) II

In a series of papers Mauduit and Sarkozy introduced measures of pseudorandomness and they constructed large families of sequences with strong pseudorandom properties. In later papers the structure of families of binary sequences was also studied. In these constructions fields with prime order were used. Throughout this paper the structure of a family of binary sequences based on GF(2 k ) will be studied.

Large Family of Sequences from Elliptic Curves over Residue Class Rings

An upper bound is established for certain exponential sums on the rational points of an elliptic curve over a residue class ring $\mathbb{Z}_N$, N=pq for two distinct odd primes p and q. The result is a generalization of an estimate of exponential sums on rational point groups of elliptic curves over finite fields. The bound is applied to showing the pseudorandomness of a large family of binary sequences constructed by using elliptic curves over $\mathbb{Z}_N$.

A Family of Binary Sequences with 4-Valued Low Correlation and Large Linear Span

For n≡0mod 4,based on d-form function,a new family of binary sequences with period 2n-1 and four-valued low correlation is proposed.The correlation distribution of the proposed family is completely determined.The linear spans of the new sequences are proved to be large and their exact values are also obtained.Compared with the known sequence families,the new family has not only low correlation,but also much larger linear spans.This family of sequences is suitable for cryptography and CDMA systems.

A Family of Binary Sequences with 4-Valued Low Correlation and Large Linear Span

A Family of Binary Sequences with 4-Valued Low Correlation and Large Linear Span

The study of collision and avalanche effect in a family of pseudorandom binary sequences

In an earlier paper we studied collisions and avalanche effect in two of the most important constructions given for large families of binary sequences possessing strong pseudorandom properties. It turned out that one of the two constructions (which is based on the use of the Legendre symbol) is ideal from this point of view, while the other construction (which is based on the size of the modulo p residue of f(n) for some polynomial f(x) ∈ \( \mathbb{F}_p \)[x]) is not satisfactory since there are “many” collisions in it. Here it is shown that this weakness of the second construction can be corrected: one can take a subfamily of the given family which is just slightly smaller and collision free.

On the complexity of a family related to the Legendre symbol

Ahlswede, Khachatrian, Mauduit and A. Sarkozy introduced the notion of family-complexity of families of binary sequences. They estimated the family-complexity of a large family related to Legendre symbol introduced by Goubin, Mauduit and Sarkozy. Here their result is improved, and apart from the constant factor the best lower bound is given for the family-complexity.

Two New Families of Optimal Binary Sequences Obtained From Quaternary Sequences

In this paper, we present two optimal binary families of sequences of length 2n-1 and 2(2n-1) for odd integer n. They are obtained as the images of proposed optimal quaternary sequences under the most significant bit and the Gray maps. The first family has 2n+1 sequences of length 2n-1 and the identical correlation distribution to that of Gold sequences and Gold-like sequences, and the second family of sequences of length 2(2n-1) has 2n sequences and the same correlation values as those of Kerdock sequences.

Construction of large families of pseudorandom binary sequences

Oon constructed large families of finite binary sequences with strong pseudorandom properties by using Dirichlet characters of large order. In this paper Oon’s construction is generalized and extended. We prove that in our construction the well-distribution and correlation measures are as “small” as in the case of the Legendre symbol.

New Families of Binary Sequences with Low Correlation and Large Size

In this paper, for odd n and any k with gcd(n, k) = 1, new binary sequence families Sk of period 2n-1 are constructed. These families have maximum correlation $1+2^{n+3\\over 2}$, family size 22n+2n+1 and maximum linear span $n(n+1)\\over 2$. The correlation distribution of Sk is completely determined as well. Compared with the modified Gold codes with the same family size, the proposed families have the same period and correlation properties, but larger linear span. As good candidates with low correlation and large family size, the new families contain the Gold sequences and the Gold-like sequences. Furthermore, Sk includes a subfamily $\\mathcal{S}^k_1$ which has the same period, correlation distribution, family size and linear span as the family So(2) recently constructed by Yu and Gong. In particular, when k=1, $\\mathcal{S}^k_1$ is exactly So(2).

Two binary sequence families with large merit factor

We calculate the asymptotic merit factor, under all rotations of sequence elements, of two families of binary sequences derived from Legendre sequences. The rotation is negaperiodic for the first family, and periodic for the second family. In both cases the maximum asymptotic merit factor is 6. As a consequence, we obtain the first two families of skew-symmetric sequences with known asymptotic merit factor, which is also 6 in both cases.

Large families of elliptic curve pseudorandom binary sequences

Large families of elliptic curve pseudorandom binary sequences

A Construction of Binary Cyclotomic Sequences Using Extension Fields

Based on the cyclotomy classes of extension fields, a family of binary cyclotomic sequences are constructed and their pseudorandom measures (i.e., the well-distribution measure and the correlation measure of order k) are estimated using certain exponential sums. A lower bound on the linear complexity profile is also presented in terms of the correlation measure.