Probabilistic Construction of Small Strongly Sum-Free Sets via Large Sidon Sets

1999 ◽  
pp. 138-143
Author(s):  
Andreas Baltz ◽  
Tomasz Schoen ◽  
Anand Srivastav
Keyword(s):  
Sidon Sets ◽  
Probabilistic Construction ◽  
Free Sets

scholarly journals Probabilistic construction of small strongly sum-free sets via large Sidon sets

2000 ◽  
Vol 86 (2) ◽  
pp. 171-176
Author(s):  
Andreas Schoen ◽  
Tomasz Srivastav ◽  
Anand Baltz
Keyword(s):  
Sidon Sets ◽  
Probabilistic Construction ◽  
Free Sets

scholarly journals On the exponents of APN power functions and Sidon sets, sum-free sets, and Dickson polynomials

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Claude Carlet ◽  
Stjepan Picek
Keyword(s):  
Necessary Conditions ◽  
Power Functions ◽  
Apn Function ◽  
Dickson Polynomials ◽  
Sidon Sets ◽  
Dickson Polynomial ◽  
Speed Up ◽  
Reversed Dickson Polynomial ◽  
Characteristic 2 ◽  
Free Sets

<p style='text-indent:20px;'>We derive necessary conditions related to the notions, in additive combinatorics, of Sidon sets and sum-free sets, on those exponents <inline-formula><tex-math id="M1">\begin{document}$ d\in {\mathbb Z}/(2^n-1){\mathbb Z} $\end{document}</tex-math></inline-formula>, which are such that <inline-formula><tex-math id="M2">\begin{document}$ F(x) = x^d $\end{document}</tex-math></inline-formula> is an APN function over <inline-formula><tex-math id="M3">\begin{document}$ {\mathbb F}_{2^n} $\end{document}</tex-math></inline-formula> (which is an important cryptographic property). We study to what extent these new conditions may speed up the search for new APN exponents <inline-formula><tex-math id="M4">\begin{document}$ d $\end{document}</tex-math></inline-formula>. We summarize all the necessary conditions that an exponent must satisfy for having a chance of being an APN, including the new conditions presented in this work. Next, we give results up to <inline-formula><tex-math id="M5">\begin{document}$ n = 48 $\end{document}</tex-math></inline-formula>, providing the number of exponents satisfying all the conditions for a function to be APN.</p><p style='text-indent:20px;'>We also show a new connection between APN exponents and Dickson polynomials: <inline-formula><tex-math id="M6">\begin{document}$ F(x) = x^d $\end{document}</tex-math></inline-formula> is APN if and only if the reciprocal polynomial of the Dickson polynomial of index <inline-formula><tex-math id="M7">\begin{document}$ d $\end{document}</tex-math></inline-formula> is an injective function from <inline-formula><tex-math id="M8">\begin{document}$ \{y\in {\Bbb F}_{2^n}^*; tr_n(y) = 0\} $\end{document}</tex-math></inline-formula> to <inline-formula><tex-math id="M9">\begin{document}$ {\Bbb F}_{2^n}\setminus \{1\} $\end{document}</tex-math></inline-formula>. This also leads to a new and simple connection between Reversed Dickson polynomials and reciprocals of Dickson polynomials in characteristic 2 (which generalizes to every characteristic thanks to a small modification): the squared Reversed Dickson polynomial of some index and the reciprocal of the Dickson polynomial of the same index are equal.</p>

Characterization of the conditional stationary distribution in Markov chains via systems of linear inequalities

2020 ◽  
Vol 52 (4) ◽  
pp. 1249-1283
Author(s):  
Masatoshi Kimura ◽  
Tetsuya Takine
Keyword(s):  
Markov Chains ◽  
Stationary Distribution ◽  
Continuous Time ◽  
State Transition ◽  
Convex Polytope ◽  
Linear Inequalities ◽  
Systems Of Linear Inequalities ◽  
Free Sets ◽  
Practical Implications

AbstractThis paper considers ergodic, continuous-time Markov chains $\{X(t)\}_{t \in (\!-\infty,\infty)}$ on $\mathbb{Z}^+=\{0,1,\ldots\}$ . For an arbitrarily fixed $N \in \mathbb{Z}^+$ , we study the conditional stationary distribution $\boldsymbol{\pi}(N)$ given the Markov chain being in $\{0,1,\ldots,N\}$ . We first characterize $\boldsymbol{\pi}(N)$ via systems of linear inequalities and identify simplices that contain $\boldsymbol{\pi}(N)$ , by examining the $(N+1) \times (N+1)$ northwest corner block of the infinitesimal generator $\textbf{\textit{Q}}$ and the subset of the first $N+1$ states whose members are directly reachable from at least one state in $\{N+1,N+2,\ldots\}$ . These results are closely related to the augmented truncation approximation (ATA), and we provide some practical implications for the ATA. Next we consider an extension of the above results, using the $(K+1) \times (K+1)$ ( $K > N$ ) northwest corner block of $\textbf{\textit{Q}}$ and the subset of the first $K+1$ states whose members are directly reachable from at least one state in $\{K+1,K+2,\ldots\}$ . Furthermore, we introduce new state transition structures called (K, N)-skip-free sets, using which we obtain the minimum convex polytope that contains $\boldsymbol{\pi}(N)$ .

scholarly journals On directed analogues of expander and hyperfinite graph sequences

2021 ◽  
pp. 1-14
Author(s):  
Endre Csóka ◽  
Łukasz Grabowski
Keyword(s):  
Directed Acyclic Graphs ◽  
Acyclic Graphs ◽  
Probabilistic Construction

Abstract We introduce and study analogues of expander and hyperfinite graph sequences in the context of directed acyclic graphs, which we call ‘extender’ and ‘hypershallow’ graph sequences, respectively. Our main result is a probabilistic construction of non-hypershallow graph sequences.

scholarly journals Colourings of the Cartesian Product of Graphs and Multiplicative Sidon Sets

2007 ◽  
Vol 28 ◽  
pp. 33-40 ◽  
Author(s):  
Attila Pór ◽  
David R. Wood
Keyword(s):  
Cartesian Product ◽  
Sidon Sets ◽  
Cartesian Product Of Graphs ◽  
Product Of Graphs

Long arithmetic progressions in sum-sets and the number x-sum-free sets

2005 ◽  
Vol 90 (02) ◽  
pp. 273-296 ◽  
Author(s):  
E. Szemerédi ◽  
V. H. Vu
Keyword(s):  
Arithmetic Progressions ◽  
Free Sets

On the Density of Universal Sum-Free Sets

1999 ◽  
Vol 8 (3) ◽  
pp. 277-280 ◽  
Author(s):  
TOMASZ SCHOEN
Keyword(s):  
Free Set ◽  
Free Sets

A set A is called universal sum-free if, for every finite 0–1 sequence χ = (e1, …, en), either(i) there exist i, j, where 1[les ]j<i[les ]n, such that ei = ej = 1 and i − j∈A, or(ii) there exists t∈N such that, for 1[les ]i[les ]n, we have t + i∈A if and only if ei = 1.It is proved that the density of each universal sum-free set is zero, which settles a problem of Cameron.

On strong Sidon sets of integers

2021 ◽  
Vol 183 ◽  
pp. 105490
Author(s):  
Yoshiharu Kohayakawa ◽  
Sang June Lee ◽  
Carlos Gustavo Moreira ◽  
Vojtěch Rödl
Keyword(s):  
Sidon Sets

Sum-free sets and short covering codes

2010 ◽  
Vol 39 (6) ◽  
Author(s):  
Emerson Carmelo ◽  
Candido Mendonça
Keyword(s):  
Covering Codes ◽  
Short Covering ◽  
Free Sets
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