Boaz Tsaban

Some New Directions in Infinite-combinatorial Topology

We give a light introduction to selection principles in topology, a young subfield of infinite-combinatorial topology. Emphasis is put on the modern approach to the problems it deals with. Recent results are described, and open problems are stated. Some results which do not appear elsewhere are also included, with proofs.

Hereditary topological diagonalizations and the Menger-Hurewicz conjectures

We consider the question of which of the major classes defined by topological diagonalizations of open or Borel covers is hereditary. Many of the classes in the open case are not hereditary already in ZFC, and none of them are provably hereditary. This is in contrast with the Borel case, where some of the classes are provably hereditary. Two of the examples are counter-examples of sizes 0 and b, respectively, to the Menger and Hurewicz Conjectures, and one of them answers a question of Steprans on perfectly meager sets.

Efficient Linear Feedback Shift Registers with Maximal Period

We introduce and analyze an efficient family of linear feedback shift registers (LFSRs) with maximal period. This family is word-oriented and is suitable for implementation in software, thus provides a solution to a recent challenge 8]. The classical theory of LFSRs is extended to provide efficient algorithms for generation of irreducible and primitive LFSRs of this new type.

Guaranteeing the diversity of number generators

A major problem in using iterative number generators of the form xi=f (xi-1) is that they can enter unexpectedly short cycles. This is hard to analyze when the generator is designed, hard to detect in real time when the generator is used, and can have devastating cryptanalytic implications. In this paper we define a measure of security, called sequence diversity, which generalizes the notion of cycle-length for noniterative generators. We then introduce the class of counter-assisted generators and show how to turn any iterative generator (even a bad one designed or seeded by an adversary) into a counter-assisted generator with a provably high diversity, without reducing the quality of generators which are already cryptographically strong. 2001 Elsevier Science

Scales, fields, and a problem of Hurewicz

Mengers basis property is a generalization of

o-BOUNDED GROUPS AND OTHER TOPOLOGICAL GROUPS WITH STRONG COMBINATORIAL PROPERTIES

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Strong γ-sets and other singular spaces

-comp\-actness and admits an elegant combinatorial interpretation. We introduce a general combinatorial method to construct non

Selection principles in mathematics: A milestone of open problems

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Polynomial-Time Solutions of Computational Problems in Noncommutative-Algebraic Cryptography

-compact sets of reals with Mengers property. Special instances of these constructions give known counterexamples to conjectures of Menger and Hurewicz. We obtain the first explicit solution to the Hurewicz 1927 problem, that was previously solved by Chaber and Pol on a dichotomic basis. The constructed sets generate nontrivial subfields of the real line with strong combinatorial properties, and most of our results can be stated in a Ramsey-theoretic manner. Since we believe that this paper is of interest to a diverse mathematical audience, we have made a special effort to make it self-contained and accessible.

Additivity properties of topological diagonalizations

We construct several topological groups with very strong combinatorial properties. In particular, we give simple examples of subgroups of R (thus strictly o-bounded) which have the Menger and Hurewicz properties but are not σ-compact, and show that the product of two o-bounded subgroups of R N may fail to be o-bounded, even when they satisfy the stronger property S 1 (B Ω ,B Ω ). This solves a problem of Tkacenko and Hernandez, and extends independent solutions of Krawczyk and Michalewski and of Banakh, Nickolas, and Sanchis. We also construct separable metrizable groups G of size continuum such that every countable Borel ω-cover of G contains a γ-cover of G.