Large families of pseudorandom binary sequences and lattices by using the multiplicative inverse

2013 ◽  
Vol 159 (2) ◽  
pp. 123-131 ◽  
Author(s):  
Huaning Liu
Keyword(s):  
Binary Sequences ◽  
Multiplicative Inverse ◽  
Large Families

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.

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>

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.)

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

Construction of pseudorandom binary sequences by using the multiplicative inverse

2005 ◽  
Vol 108 (3) ◽  
pp. 239-252 ◽  
Author(s):  
Christian Mauduit ◽  
András Sárközy
Keyword(s):  
Binary Sequences ◽  
Multiplicative Inverse

Large family of pseudorandom subsets of the set of the integers not exceeding N

2014 ◽  
Vol 10 (05) ◽  
pp. 1121-1141 ◽  
Author(s):  
Huaning Liu
Keyword(s):  
Large Family ◽  
Additive Character ◽  
Multiplicative Inverse ◽  
The Family ◽  
Large Families ◽  
Pseudorandom Measures

In a series of papers, Dartyge and Sárközy (partly with other coauthors) studied large families of pseudorandom subsets. In this paper, we introduce a new large family of pseudorandom subsets constructed by the multiplicative inverse and additive character, and study the pseudorandom measures. Furthermore, we extend the family of subsets to the case when the moduli is composite.

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