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Constructions of pseudorandom binary lattices using cyclotomic classes in finite fields

Xiaolin Chen
2020 Open Mathematics  
In this paper, we construct pseudorandom binary lattices by using cyclotomic classes in 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.  ... 
doi:10.1515/math-2020-0086 fatcat:5xqgeglnl5fcdnnn7qz3fumiv4

Counting Additive Decompositions of Quadratic Residues in Finite Fields [article]

Simon R. Blackburn, Sergei V. Konyagin, Igor E. Shparlinski
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 : aA, 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  ... 
arXiv:1403.2589v1 fatcat:cizn2rj3gbdejkvc4nqwvu2usq

Bilinear character sums over norm groups

SU HU, YAN LI
2011 Publicationes mathematicae (Debrecen)  
Let k be 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.  ... 
doi:10.5486/pmd.2011.4789 fatcat:pg7z2usrwjgtlmvm3elsgs575i

Paul Erdős's (1913-1996)

András Sárközy
1997 Acta Arithmetica  
I told him my results (which were not that exciting but, perhaps, good enough 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 < . . .  ... 
doi:10.4064/aa-81-4-299-317 fatcat:ofairefcnjar7d4gtbmcsg6l4m

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

Huaning Liu
2013 Acta Arithmetica  
In this paper we presented 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.  ... 
doi:10.4064/aa159-2-3 fatcat:uirwpwgd2nbthnc2rufct4jbwi

Additive Decompositions of Subgroups of Finite Fields [article]

Igor Shparlinski
2013 arXiv   pre-print
We say that a set S is additively decomposed into two sets A and B, if S = {a+b : aA, 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 : aA, b ∈ B}.  ... 
arXiv:1301.2872v1 fatcat:2vd3vihnbzhaxbo7yb7ozxctgy

Combinatorial and Diophantine Applications of Ergodic Theory [chapter]

Vitaly Bergelson, A. LeibmanM, Anthony Quas, Máté Wierdl
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.  ... 
doi:10.1016/s1874-575x(06)80037-8 fatcat:mcnffmdei5cizornseqkzeutra

Normal Numbers and the Normality Measure

CHRISTOPH AISTLEITNER
2013 Combinatorics, probability & computing  
The normality measure N has been introduced by Mauduit and 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.  ... 
doi:10.1017/s0963548313000084 fatcat:dbc2cninv5fp3pspoxfrbhnldy

Normal numbers and normality measure [article]

Christoph Aistleitner
2013 arXiv   pre-print
The normality measure N has been introduced by Mauduit and 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.  ... 
arXiv:1302.1919v1 fatcat:akh3qo2yjjh33lzjavb524qm5y

Construction of Large Families of Pseudorandom Subsets of the Set {1, 2, ..., N} Using Elliptic Curves

Zhixiong Chen, Li Xu, Chenhuang Wu
2010 International Journal of Network Security  
In this article, we present 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.  ... 
dblp:journals/ijnsec/ChenXW10 fatcat:qi3gv2pbhrfv3nyji4noeig6tm

ON THE SOLVABILITY OF BILINEAR EQUATIONS IN FINITE FIELDS

IGOR E. SHPARLINSKI
2008 Glasgow Mathematical Journal  
We consider the equation over 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.  ... 
doi:10.1017/s0017089508004382 fatcat:d4jx4u4if5dg5kscb7hx4gqxwq

On additive decompositions of primitive elements in finite fields [article]

Hai-Liang Wu, Yue-Feng She
2022 arXiv   pre-print
In this paper, we study several topics on additive decompositions of primitive elemements in 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.  ... 
arXiv:2202.05021v3 fatcat:geledvrdv5hehf7boqorljrrmi

On finite pseudorandom binary lattices

Katalin Gyarmati, Christian Mauduit, András Sárközy
2017 Discrete Applied Mathematics  
Here our goal is to present 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.  ... 
doi:10.1016/j.dam.2015.07.012 fatcat:i2rq7emdtvbshbnkjueqsqygjq

Density and ramsey type results on algebraic equations with restricted solution sets

Péter Csikvári, Katalin Gyarmati, András Sárközy
2012 Combinatorica  
Later Gyarmati and 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.  ... 
doi:10.1007/s00493-012-2697-9 fatcat:akeyacuc7bhjlnminpvqufu5ai

The sum of digits function in finite fields

Cécile Dartyge, András Sárközy
2013 Proceedings of the American Mathematical Society  
We define and study certain sum of digits function in the context of 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 .  ... 
doi:10.1090/s0002-9939-2013-11801-0 fatcat:turititzwvgnlhw4mm5uec6mbq
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