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Constructions of pseudorandom binary lattices using cyclotomic classes in finite fields

2020
*
Open Mathematics
*

In this paper, we construct pseudorandom binary lattices by using cyclotomic classes in

doi:10.1515/math-2020-0086
fatcat:5xqgeglnl5fcdnnn7qz3fumiv4
*finite**fields*and study the pseudorandom measure of order k, family complexity, collision and avalanche effect. ... Results indicate that such binary lattices are "good," and their families possess*a*nice structure in terms of family complexity, collision and avalanche effect. ... This is*Theorem*2 of [13] . □ Lemma 2.5. Suppose that = q p n is an odd prime power, α is*a*primitive element of the*finite**field*q and > d 1 be*a*divisor of − q 1. ...##
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Counting Additive Decompositions of Quadratic Residues in Finite Fields
[article]

2014
*
arXiv
*
pre-print

*Sárközy*has recently conjectured that the set Q of quadratic residues modulo

*a*prime p does not have nontrivial decompositions. ... We say that

*a*set S is additively decomposed into two sets

*A*and B if S = {

*a*+b :

*a*∈

*A*, b ∈ B}.

*A*. ... Acknowledgments During the preparation of the work, the second author was supported by Russian Fund

*for*Basic Research, Grant N. 14-01-00332, and Program Supporting Leading Scientific Schools, Grant Nsh ...

##
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Bilinear character sums over norm groups

2011
*
Publicationes mathematicae (Debrecen)
*

Let k be

doi:10.5486/pmd.2011.4789
fatcat:pg7z2usrwjgtlmvm3elsgs575i
*a**finite**field*with q elements. Let kn be the extension of k with degree n. ... of the norm map In this paper we estimate the bilinear character sum where U and V are arbitrary subsets of N n , ρ(U ) and θ(V ) are arbitrary bounded complex functions supported on U and V and ψ is*a*... The authors are grateful to the two anonymous referees*for*their valuable comments. ...##
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Paul Erdős's (1913-1996)

1997
*
Acta Arithmetica
*

I told him my results (which were not that exciting but, perhaps, good enough

doi:10.4064/aa-81-4-299-317
fatcat:ofairefcnjar7d4gtbmcsg6l4m
*for**a*beginner) and I sketched the proofs. ... FAREWELL, PAUL I was 19 years old,*a*second year university student, when I received the following letter: "Dear Mr.*Sárközy*, I have heard about your nice results" so and so "from Paul Turán. ... Erdős,*Sárközy*and Szemerédi [67.04, 67.05] also sharpened Behrend's*theorem*on the estimate of*a*i ≤x 1/*a*i*for*primitive sequences*a*1 <*a*2 < . . . ...##
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Large families of pseudorandom binary sequences and lattices by using the multiplicative inverse

2013
*
Acta Arithmetica
*

In this paper we presented

doi:10.4064/aa159-2-3
fatcat:uirwpwgd2nbthnc2rufct4jbwi
*a*few large families of pseudorandom binary sequences and lattices, and generalized some existed constructions. ... Pseudorandom binary sequences and lattices play an important role in cryptography, so in*a*series of papers many sequences and lattices have been given and studied. ... Let = n q p and q be*a**finite**field*, and let v 1 , …, v n be linearly independent elements of q over p . C. Mauduit and*A*.*Sárközy*[19] presented the following construction. ...##
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Additive Decompositions of Subgroups of Finite Fields
[article]

2013
*
arXiv
*
pre-print

We say that

arXiv:1301.2872v1
fatcat:2vd3vihnbzhaxbo7yb7ozxctgy
*a*set S is additively decomposed into two sets*A*and B, if S = {*a*+b :*a*∈*A*, b ∈ B}. Here we study additively decompositions of multiplicative subgroups of*finite**fields*. ... Dartyge and*A*.*Sarkozy*on additive decompositions of quadratic residues and primitive roots modulo p. ... Introduction Let F q be the*finite**field*of q elements. As usual,*for*two sets*A*, B ⊆ F q we define their sum as*A*+ B = {*a*+ b :*a*∈*A*, b ∈ B}. ...##
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Combinatorial and Diophantine Applications of Ergodic Theory
[chapter]

2006
*
Handbook of Dynamical Systems
*

*For*fixed n ∈ N and

*a*large enough prime p, the polynomial f (z, y) = z n − y n represents the

*finite*

*field*Z p = Z/pZ. ... Fermat's

*theorem*over

*finite*

*fields*Our first example is related to Fermat's last

*theorem*. ...

##
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Normal Numbers and the Normality Measure

2013
*
Combinatorics, probability & computing
*

The normality measure N has been introduced by Mauduit and

doi:10.1017/s0963548313000084
fatcat:dbc2cninv5fp3pspoxfrbhnldy
*Sárközy*in order to describe the pseudorandomness properties of*finite*binary sequences. ... In the present paper we improve the upper bound to c(log N ) 2*for*some constant c, by this means solving the problem of the asymptotic order of the minimal value of the normality measure up to*a*logarithmic ... Programming, and the normality measure of Mauduit and*Sárközy*is*a*quantitative version of such*a*pseudorandomness test*for*the case of*a**finite*sequence of digits. ...##
###
Normal numbers and normality measure
[article]

2013
*
arXiv
*
pre-print

The normality measure N has been introduced by Mauduit and

arXiv:1302.1919v1
fatcat:akh3qo2yjjh33lzjavb524qm5y
*Sárközy*in order to describe the pseudorandomness properties of*finite*binary sequences. ... In the present paper we improve the upper bound to c ( N)^2*for*some constant c, by this means solving the problem of the asymptotic order of the minimal value of the normality measure up to*a*logarithmic ... Programming, and the normality measure of Mauduit and*Sárközy*is*a*quantitative version of such*a*pseudorandomness test*for*the case of*a**finite*sequence of digits. ...##
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Construction of Large Families of Pseudorandom Subsets of the Set {1, 2, ..., N} Using Elliptic Curves

2010
*
International Journal of Network Security
*

In this article, we present

dblp:journals/ijnsec/ChenXW10
fatcat:qi3gv2pbhrfv3nyji4noeig6tm
*a*construction of pseudorandom subsets by using elliptic curves over*finite**fields*and estimate their pseudorandom measures. ... several constructive examples*for*subsets with strong pseudorandom properties when N is*a*prime number. ... It is*a*natural way to choose elliptic curves over*finite**fields*, partially*for*the elliptic curve cryptography*for*extensive use. ...##
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ON THE SOLVABILITY OF BILINEAR EQUATIONS IN FINITE FIELDS

2008
*
Glasgow Mathematical Journal
*

We consider the equation over

doi:10.1017/s0017089508004382
fatcat:d4jx4u4if5dg5kscb7hx4gqxwq
*a**finite**field*ކ q of q elements, with variables from arbitrary sets*A*, B, C, D ⊆ ކ q . ... above equation has*a*solution*for*any λ ∈ ކ * q . ... The author is grateful to Andras*Sárközy**for*several useful comments. This work was supported in part by ARC grant DP0556431. ...##
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On additive decompositions of primitive elements in finite fields
[article]

2022
*
arXiv
*
pre-print

In this paper, we study several topics on additive decompositions of primitive elemements in

arXiv:2202.05021v3
fatcat:geledvrdv5hehf7boqorljrrmi
*finite**fields*. Also we refine some bounds obtained by Dartyge and*Sárközy*and Shparlinski. ... Hao Pan*for*his encouragement. ... Introduction Let p be an odd prime and let F p be the*finite**field*of p elements. Let F × p be the set of all non-zero elements of F p . It is known that F × p is*a*cyclic group. ...##
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On finite pseudorandom binary lattices

2017
*
Discrete Applied Mathematics
*

Here our goal is to present

doi:10.1016/j.dam.2015.07.012
fatcat:i2rq7emdtvbshbnkjueqsqygjq
*a*survey of all these papers. 2010 Mathematics Subject Classification: Primary 11K45. ... This extension started with*a*paper published in 2006, and since that about 25 papers have been written on this subject. ... In [26] Liu presented another construction*for**a*large family of pseudorandom binary lattices by using the multiplicative inverse and the quadratic character of*finite**fields*. ...##
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Density and ramsey type results on algebraic equations with restricted solution sets

2012
*
Combinatorica
*

Later Gyarmati and

doi:10.1007/s00493-012-2697-9
fatcat:akeyacuc7bhjlnminpvqufu5ai
*Sárközy*generalized and extended these problems by studying these equations and also other algebraic equations with restricted solution sets over*finite**fields*. ... In earlier papers*Sárközy*studied the solvability of the equations where*A*, B, C, D are "large" subsets of F p . ... Gyarmati and*Sárközy*[6] , [7] generalized these results to*finite**fields*:*Theorem**A*If q is*a*prime power,*A*, B, C, D ⊂ F q and |*A*| |B| |C| |D| > q 3 , then (1.1) can be solved. ...##
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The sum of digits function in finite fields

2013
*
Proceedings of the American Mathematical Society
*

We define and study certain sum of digits function in the context of

doi:10.1090/s0002-9939-2013-11801-0
fatcat:turititzwvgnlhw4mm5uec6mbq
*finite**fields*. We give the number of polynomial values of F q with*a*fixed sum of digits. ... We also state*a*result*for*the sum of digits of polynomial values with generator arguments. ... In this paper our goal is to study the analogs of some of these problems in*finite**fields*. Indeed, let p be*a*prime number, q = p r with r > 2, and consider the*field*F q . ...
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