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 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.

scholarly journals Designing an Efficient and Highly Dynamic Substitution-Box Generator for Block Ciphers Based on Finite Elliptic Curves

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Ghulam Murtaza ◽  
Naveed Ahmed Azam ◽  
Umar Hayat
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Computation Time ◽  
Binary Sequences ◽  
Algebraic Complexity ◽  
Computational Results ◽  
Computational Power ◽  
Substitution Box ◽  
Differential Approximation ◽  
Lightweight Cryptography

Developing a substitution-box (S-box) generator that can efficiently generate a highly dynamic S-box with good cryptographic properties is a hot topic in the field of cryptography. Recently, elliptic curve (EC)-based S-box generators have shown promising results. However, these generators use large ECs to generate highly dynamic S-boxes and thus may not be suitable for lightweight cryptography, where the computational power is limited. The aim of this paper is to develop and implement such an S-box generator that can be used in lightweight cryptography and perform better in terms of computation time and security resistance than recently designed S-box generators. To achieve this goal, we use ordered ECs of small size and binary sequences to generate certain sequences of integers which are then used to generate S-boxes. We performed several standard analyses to test the efficiency of the proposed generator. On an average, the proposed generator can generate an S-box in 0.003 seconds, and from 20,000 S-boxes generated by the proposed generator, 93 % S-boxes have at least the nonlinearity 96. The linear approximation probability of 1000 S-boxes that have the best nonlinearity is in the range [0.117, 0.172] and more than 99% S-boxes have algebraic complexity at least 251. All these S-boxes have the differential approximation probability value in the interval [0.039, 0.063]. Computational results and comparisons suggest that our newly developed generator takes less running time and has high security against modern attacks as compared to several existing well-known generators, and hence, our generator is suitable for lightweight cryptography. Furthermore, the usage of binary sequences in our generator allows generating plaintext-dependent S-boxes which is crucial to resist chosen-plaintext attacks.

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 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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