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

2012 ◽  
Vol 154 (1) ◽  
pp. 103-108 ◽  
Author(s):  
Huaning Liu ◽  
Jing Gao
Keyword(s):  
Binary Sequences ◽  
Legendre Symbol ◽  
Large Families

scholarly journals Quasi-Random Graphs, Pseudo-Random Graphs and Pseudorandom Binary Sequences, I. (Quasi-Random Graphs)

2019 ◽  
Vol 14 (2) ◽  
pp. 103-126
Author(s):  
József Borbély ◽  
András Sárközy
Keyword(s):  
Random Graphs ◽  
Adjacency Matrix ◽  
Binary Sequence ◽  
Binary Sequences ◽  
Circulant Graph ◽  
Large Families

AbstractIn the last decades many results have been proved on pseudo-randomness of binary sequences. In this series our goal is to show that using many of these results one can also construct large families of quasi-random, pseudo-random and strongly pseudo-random graphs. Indeed, it will be proved that if the first row of the adjacency matrix of a circulant graph forms a binary sequence which possesses certain pseudorandom properties (and there are many large families of binary sequences known with these properties), then the graph is quasi-random, pseudo-random or strongly pseudo-random, respectively. In particular, here in Part I we will construct large families of quasi-random graphs along these lines. (In Parts II and III we will present and study constructions for pseudo-random and strongly pseudo-random graphs, respectively.)

GOWERS UNIFORMITY NORM AND PSEUDORANDOM MEASURES OF THE PSEUDORANDOM BINARY SEQUENCES

2011 ◽  
Vol 07 (05) ◽  
pp. 1279-1302 ◽  
Author(s):  
HUANING LIU
Keyword(s):  
Exponential Sums ◽  
Binary Sequences ◽  
Vandermonde Determinant ◽  
Additive Character ◽  
Pseudorandom Sequences ◽  
Multiplicative Inverse ◽  
Large Families ◽  
Points Of View ◽  
Pseudorandom Measures ◽  
Gowers Norm

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.

Concatenation of Legendre symbol sequences

2011 ◽  
Vol 48 (2) ◽  
pp. 193-204
Author(s):  
Katalin Gyarmati
Keyword(s):  
Binary Sequence ◽  
Binary Sequences ◽  
Legendre Symbol ◽  
Pseudorandom Binary Sequence ◽  
Symbol Sequences

In the applications it may occur that our initial pseudorandom binary sequence is not long enough, thus we have to take the concatenation of it with another pseudorandom binary sequences. Here we will consider concatenation of Legendre symbol sequences so that the resulting longer sequence has strong pseudorandom properties.

scholarly journals Large families of elliptic curve pseudorandom binary sequences

2009 ◽  
Vol 140 (2) ◽  
pp. 135-144 ◽  
Author(s):  
Huaning Liu ◽  
Tao Zhan ◽  
Xiaoyun Wang
Keyword(s):  
Elliptic Curve ◽  
Binary Sequences ◽  
Large Families

scholarly journals Binary sequences and lattices constructed by discrete logarithms

2021 ◽  
Vol 7 (3) ◽  
pp. 4655-4671
Author(s):  
Yuchan Qi ◽  
◽  
Huaning Liu
Keyword(s):  
Exponential Sums ◽  
Binary Sequence ◽  
Large Family ◽  
Binary Sequences ◽  
Multiplicative Inverse ◽  
Discrete Logarithms ◽  
Modulo P ◽  
Large Families ◽  
Definition Of ◽  
Pseudorandom Measures

<abstract><p>In 1997, Mauduit and Sárközy first introduced the measures of pseudorandomness for binary sequences. Since then, many pseudorandom binary sequences have been constructed and studied. In particular, Gyarmati presented a large family of pseudorandom binary sequences using the discrete logarithms. Ten years later, to satisfy the requirement from many applications in cryptography (e.g., in encrypting "bit-maps'' and watermarking), the definition of binary sequences is extended from one dimension to several dimensions by Hubert, Mauduit and Sárközy. They introduced the measure of pseudorandomness for this kind of several-dimension binary sequence which is called binary lattices. In this paper, large families of pseudorandom binary sequences and binary lattices are constructed by both discrete logarithms and multiplicative inverse modulo $ p $. The upper estimates of their pseudorandom measures are based on estimates of either character sums or mixed exponential sums.</p></abstract>

Linear Complexity of Binary Threshold Sequences Derived from Generalized Polynomial Quotient with Prime-Power Modulus

2020 ◽  
Vol 31 (05) ◽  
pp. 569-581
Author(s):  
Lianhua Wang ◽  
Xiaoni Du
Keyword(s):  
Prime Power ◽  
Linear Complexity ◽  
Binary Sequences ◽  
Legendre Symbol ◽  
Complexity Results ◽  
Generalized Polynomial

In this paper, firstly we extend the polynomial quotient modulo an odd prime [Formula: see text] to its general case with modulo [Formula: see text] and [Formula: see text]. From the new quotient proposed, we define a class of [Formula: see text]-periodic binary threshold sequences. Then combining the Legendre symbol and Euler quotient modulo [Formula: see text] together, with the condition of [Formula: see text], we present exact values of the linear complexity for [Formula: see text], and all the possible values of the linear complexity for [Formula: see text]. The linear complexity is very close to the period and is of desired value for cryptographic purpose. Our results extend the linear complexity results of the corresponding [Formula: see text]-periodic binary sequences in earlier work.

scholarly journals On finite pseudorandom binary sequences I: Measure of pseudorandomness, the Legendre symbol

1997 ◽  
Vol 82 (4) ◽  
pp. 365-377 ◽  
Author(s):  
Christian Mauduit ◽  
András Sárközy
Keyword(s):  
Binary Sequences ◽  
Legendre Symbol

scholarly journals Construction of large families of pseudorandom binary sequences

2004 ◽  
Vol 106 (1) ◽  
pp. 56-69 ◽  
Author(s):  
Louis Goubin ◽  
Christian Mauduit ◽  
András Sárközy
Keyword(s):  
Binary Sequences ◽  
Large Families
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