Random Sidon Sequences

Archive for Mathematical Logic, 2001

Let ω be a Kolmogorov-Chaitin random sequence with ω 1:n denoting the first n digits of ω. Let P be a recursive predicate defined on all finite binary strings such that the Lebesgue measure of the set {ω|∃nP (ω 1:n)} is a computable real α. Roughly, P holds with computable probability for a random infinite sequence. Then there is an algorithm which on input indices for any such P and α finds an n such that P holds within the first n digits of ω or not in ω at all. We apply the result to the halting probability and show that various generalizations of the result fail.

Russian Mathematical Surveys. 45:1, 1990

How can one define that an individual binary sequence is random? The paper presents a comparative study of different approaches to this problem. The three properties of randomness to be analysed are: stochastic, chaotic, and typic (i.e., random in the senses given by von Mises, Kolmogorov, and Martin-Löf, respectively). The randomness of a (binary) sequence is to be considered with respect to some probability distribution over the set of all the sequences, e.g., the uniform Bernoulli distribution. The concept underlying the typicness is that of effectively zero sets of sequences: a sequence is said to be typic if it is not contained in any effectively zero set of sequences. The concept underlying the chaoticness is that of monotone entropy (complexity) KM(x) of any given finite sequence x: a sequence is said to be chaotic if KM(x)=−logP(Ωx)+O(1) for any initial segment x of this sequence. (Ωx is the set of all infinite sequences initiated by x, and P(Ωx) is the measure of this set in the given probability distribution.) The Levin-Schnorr theorem shows that a sequence is typic (w.r.t. the uniform Bernoulli distribution) iff it is chaotic. A sequence x is stochastic (i.e., random in the von Mises’ sense) w.r.t the uniform Bernoulli distribution if for any infinite subsequence of x chosen by some “admissible rule of choice”, the frequency of occurrence of the symbol 1 in initial segments of this subsequence tends to 1/2 as the lengths of these segments increase infinitely. Here the notion “admissible rule of choice” is requested to be defined. Two definitions of this notion introduced by Church and by Kolmogorov-Loveland are considered, and, according to them two definitions of the notion “stochastic sequence” are obtained. Unfortunately, these two definitions are not equivalent, and every of them is weaker than that of a typic, or chaotic, sequence.

Arkiv för matematik, 1993

Let B denote an arbitrary Banach space, G a compact abelian group with Haar measure µ and dual group Γ. Let E be a Sidon subset of Γ with Sidon constant S(E). Let r n denote the n-th Rademacher function on [0, 1]. We show that there is a constant c, depending only on S(E), such that, for all α > 0:

arXiv (Cornell University), 2021

For a given positive integer k, the Sidon-Ramsey number SR(k) is defined as the minimum value of n such that, in every partition of the set [1, n] into k parts, there exists a part that contains two distinct pairs of numbers with the same sum. In other words, there is a part that is not a Sidon set. In this paper, we investigate the asymptotic behavior of this parameter and two generalizations of it. The first generalization involves replacing pairs of numbers with h-tuples, such that in every partition of [1, n] into k parts, there exists a part that contains two distinct h-tuples with the same sum. Alternatively, there is a part that is not a B h set. The second generalization considers the scenario where the interval [1, n] is substituted with a nonnecessarily symmetric d-dimensional box of the form d i=1 [1, n i ]. For the general case of h ≥ 3 and non-symmetric boxes, before applying our method to obtain the Ramsey-type result, we needed to establish an upper bound for the corresponding density parameter.

Lecture Notes in Computer Science, 2008

Cryptography and Communications, 2017

We study the pseudorandomness of automatic sequences in terms of well-distribution and correlation measure of order 2. We detect non-random behavior which can be derived either from the functional equations satisfied by their generating functions or from their generating finite automatons, respectively.

Information and Control, 1966

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Cryptologia, 2020

In the ElGamal signature and encryption schemes, an element x of the underlying group G = Z × p = {1, . . . , p -1} for a prime p is also considered as an exponent, for example in g x , where g is a generator of G. This ElGamal map x → g x is poorly understood, and one may wonder whether it has some randomness properties. The underlying map from G to Zp-1 with x → x is trivial from a computer science point of view, but does not seem to have any mathematical structure. This work presents two pieces of evidence for randomness. Firstly, experiments with small primes suggest that the map behaves like a uniformly random permutation with respect to two properties that we consider. Secondly, the theory of Sidon sets shows that the graph of this map is equidistributed in a suitable sense. It remains an open question to prove more randomness properties, for example, that the ElGamal map is pseudorandom.

Combinatorics, Probability and Computing, 2006

Mauduit and Sárközy introduced and studied certain numerical parameters associated to finite binary sequences EN ∈ {−1, 1} N in order to measure their 'level of randomness'. Those parameters, the normality measure N (EN ), the well-distribution measure W (EN ), and the correlation measure C k (EN ) of order k, focus on different combinatorial aspects of EN . In their work, amongst others, Mauduit and Sárközy (i) investigated the relationship among those parameters and their minimal possible value, (ii) estimated N (EN ), W (EN ), and C k (EN ) for certain explicitly constructed sequences EN suggested to have a 'pseudorandom nature', and (iii) investigated the value of those parameters for genuinely random sequences EN .

2011 IEEE International Symposium on Information Theory Proceedings, 2011

Sidon sequences and their generalizations have found during the years and especially recently various applications in coding theory. One of the most important applications of these sequences is in the connection of synchronization patterns. A few constructions of two-dimensional synchronization patterns are based on these sequences. In this paper we present sufficient conditions that a two-dimensional synchronization pattern can be transformed into a Sidon sequence. We also present a new construction for Sidon sequences over an alphabet of size q(q -1), where q is a power of a prime.