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>

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.

Large families of pseudorandom binary lattices by using the multiplicative inverse modulo P

2019 ◽  
Vol 15 (03) ◽  
pp. 527-546
Author(s):  
Huaning Liu
Keyword(s):  
Exponential Sums ◽  
Character Sums ◽  
Two Dimensional ◽  
Multiplicative Inverse ◽  
Avalanche Effect ◽  
One Dimensional ◽  
Modulo P ◽  
Large Families ◽  
Pseudorandom Measures

Hubert, Mauduit and Sárközy introduced pseudorandom measures for finite pseudorandom binary lattices. Gyarmati, Mauduit, Sárközy and Stewart presented some natural and flexible constructions, which are the two-dimensional extensions and modifications of a few one-dimensional constructions. The upper estimates for the pseudorandom measures of their binary lattices are based on the principle that character sums or exponential sums in two variables can be estimated by fixing one of the variables. In this paper, we constructed two large families of [Formula: see text] dimensional pseudorandom binary lattices by using the multiplicative inverse modulo [Formula: see text], and study the properties: pseudorandom measure, collision and avalanche effect.

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.

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 Topics in Algorithmic Randomness and Computability Theory

2021 ◽  
Author(s):  
◽  
Michael McInerney
Keyword(s):  
Binary Sequence ◽  
Computability Theory ◽  
Binary Sequences ◽  
Random Sets ◽  
Turing Degrees ◽  
Algorithmic Randomness ◽  
Computational Power ◽  
Definition Of ◽  
Compare And Contrast

<p>This thesis establishes results in several different areas of computability theory.  The first chapter is concerned with algorithmic randomness. A well-known approach to the definition of a random infinite binary sequence is via effective betting strategies. A betting strategy is called integer-valued if it can bet only in integer amounts. We consider integer-valued random sets, which are infinite binary sequences such that no effective integer-valued betting strategy wins arbitrarily much money betting on the bits of the sequence. This is a notion that is much weaker than those normally considered in algorithmic randomness. It is sufficiently weak to allow interesting interactions with topics from classical computability theory, such as genericity and the computably enumerable degrees. We investigate the computational power of the integer-valued random sets in terms of standard notions from computability theory.  In the second chapter we extend the technique of forcing with bushy trees. We use this to construct an increasing ѡ-sequence 〈an〉 of Turing degrees which forms an initial segment of the Turing degrees, and such that each an₊₁ is diagonally noncomputable relative to an. This shows that the DNR₀ principle of reverse mathematics does not imply the existence of Turing incomparable degrees.   In the final chapter, we introduce a new notion of genericity which we call ѡ-change genericity. This lies in between the well-studied notions of 1- and 2-genericity. We give several results about the computational power required to compute these generics, as well as other results which compare and contrast their behaviour with that of 1-generics.</p>

scholarly journals Topics in Algorithmic Randomness and Computability Theory

2021 ◽  
Author(s):  
◽  
Michael McInerney
Keyword(s):  
Binary Sequence ◽  
Computability Theory ◽  
Binary Sequences ◽  
Random Sets ◽  
Turing Degrees ◽  
Algorithmic Randomness ◽  
Computational Power ◽  
Definition Of ◽  
Compare And Contrast

<p>This thesis establishes results in several different areas of computability theory.  The first chapter is concerned with algorithmic randomness. A well-known approach to the definition of a random infinite binary sequence is via effective betting strategies. A betting strategy is called integer-valued if it can bet only in integer amounts. We consider integer-valued random sets, which are infinite binary sequences such that no effective integer-valued betting strategy wins arbitrarily much money betting on the bits of the sequence. This is a notion that is much weaker than those normally considered in algorithmic randomness. It is sufficiently weak to allow interesting interactions with topics from classical computability theory, such as genericity and the computably enumerable degrees. We investigate the computational power of the integer-valued random sets in terms of standard notions from computability theory.  In the second chapter we extend the technique of forcing with bushy trees. We use this to construct an increasing ѡ-sequence 〈an〉 of Turing degrees which forms an initial segment of the Turing degrees, and such that each an₊₁ is diagonally noncomputable relative to an. This shows that the DNR₀ principle of reverse mathematics does not imply the existence of Turing incomparable degrees.   In the final chapter, we introduce a new notion of genericity which we call ѡ-change genericity. This lies in between the well-studied notions of 1- and 2-genericity. We give several results about the computational power required to compute these generics, as well as other results which compare and contrast their behaviour with that of 1-generics.</p>

scholarly journals Two tonoplast MATE proteins function as turgor-regulating chloride channels inArabidopsis

2017 ◽  
Vol 114 (10) ◽  
pp. E2036-E2045 ◽  
Author(s):  
Haiwen Zhang ◽  
Fu-Geng Zhao ◽  
Ren-Jie Tang ◽  
Yuexuan Yu ◽  
Jiali Song ◽  
...  
Keyword(s):  
Chloride Channels ◽  
Ectopic Expression ◽  
Root Hairs ◽  
Large Family ◽  
Patch Clamp Recording ◽  
Mesophyll Cells ◽  
Environmental Signals ◽  
Turgor Regulation ◽  
Cell Turgor ◽  
Definition Of

The central vacuole in a plant cell occupies the majority of the cellular volume and plays a key role in turgor regulation. The vacuolar membrane (tonoplast) contains a large number of transporters that mediate fluxes of solutes and water, thereby adjusting cell turgor in response to developmental and environmental signals. We report that two tonoplast Detoxification efflux carrier (DTX)/Multidrug and Toxic Compound Extrusion (MATE) transporters, DTX33 and DTX35, function as chloride channels essential for turgor regulation inArabidopsis. Ectopic expression of each transporter inNicotiana benthamianamesophyll cells elicited a large voltage-dependent inward chloride current across the tonoplast, showing that DTX33 and DTX35 each constitute a functional channel. Both channels are highly expressed inArabidopsistissues, including root hairs and guard cells that experience rapid turgor changes during root-hair elongation and stomatal movements. Disruption of these two genes, either in single or double mutants, resulted in shorter root hairs and smaller stomatal aperture, with double mutants showing more severe defects, suggesting that these two channels function additively to facilitate anion influx into the vacuole during cell expansion. In addition,dtx35single mutant showed lower fertility as a result of a defect in pollen-tube growth. Indeed, patch-clamp recording of isolated vacuoles indicated that the inward chloride channel activity across the tonoplast was impaired in the double mutant. Because MATE proteins are widely known transporters of organic compounds, finding MATE members as chloride channels expands the functional definition of this large family of transporters.

scholarly journals Effects of the population pedigree on genetic signatures of historical demographic events

2016 ◽  
Vol 113 (29) ◽  
pp. 7994-8001 ◽  
Author(s):  
John Wakeley ◽  
Léandra King ◽  
Peter R. Wilton
Keyword(s):  
Population Genetics ◽  
Selective Sweep ◽  
Large Family ◽  
Recent Past ◽  
Structure Population ◽  
Genetic Transmission ◽  
Large Samples ◽  
Large Families ◽  
Genetic Signatures ◽  
Unique Signature

Genetic variation among loci in the genomes of diploid biparental organisms is the result of mutation and genetic transmission through the genealogy, or population pedigree, of the species. We explore the consequences of this for patterns of variation at unlinked loci for two kinds of demographic events: the occurrence of a very large family or a strong selective sweep that occurred in the recent past. The results indicate that only rather extreme versions of such events can be expected to structure population pedigrees in such a way that unlinked loci will show deviations from the standard predictions of population genetics, which average over population pedigrees. The results also suggest that large samples of individuals and loci increase the chance of picking up signatures of these events, and that very large families may have a unique signature in terms of sample distributions of mutant alleles.

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