arXiv subjects

Boaz Tsaban

Publications and source records attributed to Boaz Tsaban.

At least 55 records · Page 3Linked to original sources

arXiv2012

The Discrete Logarithm Problem in Bergman's non-representable ring

Bergman's Ring $E_p$, parameterized by a prime number $p$, is a ring with $p^5$ elements that cannot be embedded in a ring of matrices over any commutative ring. This ring was discovered in 1974. In 2011, Climent, Navarro and Tortosa described an efficient implementation of $E_p$ using simple modular arithmetic, and suggested that this ring may be a useful source for intractable cryptographic problems. We present a deterministic polynomial time reduction of the Discrete Logarithm Problem in $E_p$ to the classical Discrete Logarithm Problem in $\Zp$, the $p$-element field. In particular, the Discrete Logarithm Problem in $E_p$ can be solved, by conventional computers, in sub-exponential time.

cs.CR↗

arXiv2012

SPM Bulletin 34

This is the 34th issue of this bulletin, dedicated to selection principles in mathematics. Announcements include, among other things, a call for papers for the Topology and its Applications special issue on selection principles, and the solution of the minimal tower problem in a surprising way.

math.GN↗

arXiv2012

Diagonalizations of dense families

We develop a unified framework for the study of properties involving diagonalizations of dense families in topological spaces. We provide complete classification of these properties. Our classification draws upon a large number of methods and constructions scattered in the literature, and on some novel results concerning the classical properties.

math.GN↗

arXiv2012

Additivity of the Gerlits--Nagy property and concentrated sets

We settle all problems posed by Scheepers, in his tribute paper to Gerlits, concerning the additivity of the Gerlits--Nagy property and related additivity numbers. We apply these results to compute the minimal number of concentrated sets of reals (in the sense of Besicovitch) whose union, when multiplied with a Gerlits--Nagy space, need not have Rothberger's property. We apply these methods to construct a large family of spaces, whose product with every Hurewicz space has Menger's property.

math.GN↗

arXiv2012

On the cardinality of the $θ$-closed hull of sets

The θ-closed hull of a set A in a topological space is the smallest set C containing A such that, whenever all $closed$ neighborhoods of a point intersect C, this point is in C. We define a new topological cardinal invariant function, the $θ-bitighness small number$ of a space X, bts_theta(X), and prove that in every topological space X, the cardinality of the theta-closed hull of each set A is at most |A|^{bts_theta(X)}. Using this result, we synthesize all earlier results on bounds on the cardinality of theta-closed hulls. We provide applications to P-spaces and to the almost-Lindelof number.

math.GN↗

arXiv2012

Short expressions of permutations as products and cryptanalysis of the Algebraic Eraser

On March 2004, Anshel, Anshel, Goldfeld, and Lemieux introduced the \emph{Algebraic Eraser} scheme for key agreement over an insecure channel, using a novel hybrid of infinite and finite noncommutative groups. They also introduced the \emph{Colored Burau Key Agreement Protocol (CBKAP)}, a concrete realization of this scheme. We present general, efficient heuristic algorithms, which extract the shared key out of the public information provided by CBKAP. These algorithms are, according to heuristic reasoning and according to massive experiments, successful for all sizes of the security parameters, assuming that the keys are chosen with standard distributions. Our methods come from probabilistic group theory (permutation group actions and expander graphs). In particular, we provide a simple algorithm for finding short expressions of permutations in $S_n$, as products of given random permutations. Heuristically, our algorithm gives expressions of length $O(n^2\log n)$, in time and space $O(n^3)$. Moreover, this is provable from \emph{the Minimal Cycle Conjecture}, a simply stated hypothesis concerning the uniform distribution on $S_n$. Experiments show that the constants in these estimations are small. This is the first practical algorithm for this problem for $n\ge 256$. Remark: \emph{Algebraic Eraser} is a trademark of SecureRF. The variant of CBKAP actually implemented by SecureRF uses proprietary distributions, and thus our results do not imply its vulnerability. See also arXiv:abs/12020598

math.GR↗

arXiv2012

SPM Bulletin 33

33rd issue of a bulletin dedicated to research on selective properties in mathematics.

math.GN↗

arXiv2012

Hereditarily Hurewicz spaces and Arhangel'skii sheaf amalgamations

A classical theorem of Hurewicz characterizes spaces with the Hurewicz covering property as those having bounded continuous images in the Baire space. We give a similar characterization for spaces X which have the Hurewicz property hereditarily. We proceed to consider the class of Arhangel'skii alpha_1 spaces, for which every sheaf at a point can be amalgamated in a natural way. Let C_p(X) denote the space of continuous real-valued functions on X with the topology of pointwise convergence. Our main result is that C_p(X) is an alpha_1 space if, and only if, each Borel image of X in the Baire space is bounded. Using this characterization, we solve a variety of problems posed in the literature concerning spaces of continuous functions.

math.GN↗

arXiv2011

Selection Principles and special sets of reals: Open problems

We give a selection of major open problems involving selective properties, diagonalizations, and covering properties for sets of real numbers. This is a revision of the version published as a chapter in the book \textbf{Open Problems in Topology II} (E. Pearl, ed.), Elsevier B.V., 2007, 91--108. The present version reports solutions of some problems, uses up-to-date notation, and update bibliography. Comments and further updates would be appreciated.

math.GN↗

arXiv2011

SPM Bulletin 32

In addition to announcements of several new papers, this issue contains a brief personal memorandum for Misha Matveev. The issue also announces the coming SPM meeting (June 2012).

math.GN↗

arXiv2011

Superfilters, Ramsey theory, and van der Waerden's Theorem

Superfilters are generalized ultrafilters, which capture the underlying concept in Ramsey theoretic theorems such as van der Waerden's Theorem. We establish several properties of superfilters, which generalize both Ramsey's Theorem and its variant for ultrafilters on the natural numbers. We use them to confirm a conjecture of Kočinac and Di Maio, which is a generalization of a Ramsey theoretic result of Scheepers, concerning selections from open covers. Following Bergelson and Hindman's 1989 Theorem, we present a new simultaneous generalization of the theorems of Ramsey, van der Waerden, Schur, Folkman-Rado-Sanders, Rado, and others, where the colored sets can be much smaller than the full set of natural numbers.

math.CO↗

arXiv2011

SPM Bulletin 18

CONTENTS: A surprising covering of the real line Unions of chains in dyadic compact spaces and topological groups On the Pytkeev property in spaces of continuous functions Selection principles related to alpha_i-properties On the Kocinac alpha_i properties A new selection principle First Countable Continua and Proper Forcing The convergence space of minimal usco mappings D-forced spaces: a new approach to resolvability Resolvability of spaces having small spread or extent Resolvability and monotone normality Isomorphism of Borel full groups A Poset Hierarchy Infinite asymptotic games Elementary submodels and separable monotonically normal compacta An application of CAT A general Stone representation theorem Measure Recognition Problem Cardinal invariants for C-cross topologies Problem of the Issue.

math.GN↗

arXiv2011

Continuous selections and sigma-spaces

Assume that X is a metrizable separable space, and each clopen-valued lower semicontinuous multivalued map Phi from X to Q has a continuous selection. Our main result is that in this case, X is a sigma-space. We also derive a partial converse implication, and present a reformulation of the Scheepers Conjecture in the language of continuous selections.

math.GN↗

arXiv2011

Hurewicz sets of reals without perfect subsets

We show that even for subsets X of the real line which do not contain perfect sets, the Hurewicz property does not imply the property S1(Gamma,Gamma), asserting that for each countable family of open gamma-covers of X, there is a choice function whose image is a gamma-cover of X. This settles a problem of Just, Miller, Scheepers, and Szeptycki. Our main result also answers a question of Bartoszynski and Tsaban, and implies that for C_p(X), the conjunction of Sakai's strong countable fan tightness and the Reznichenko property does not imply Arhangelskii's property alpha_2.

math.GN↗

arXiv2011

SPM Bulletin 31

Among the many papers announced here, a recent series of papers of Franklin Tall on selective properties (SPM) is noteworthy.

math.GN↗

arXiv2011

On productively Lindelöf spaces

The class of spaces such that their product with every Lindelöf space is Lindelöf is not well-understood. We prove a number of new results concerning such productively Lindelöf spaces with some extra property, mainly assuming the Continuum Hypothesis.

math.GN↗

arXiv2010

On a problem of Juhasz and van Mill

A 27 years old and still open problem of Juhasz and van Mill asks whether there exists a cardinal kappa such that every regular dense in itself countably compact space has a dense in itself subset of cardinality at most kappa. We give a negative answer for the analogous question where_regular_ is weakened to_Hausdorff_, and_coutnably compact_ is strengthened to_sequentially compact_.

math.GN↗

arXiv2010

On the Pytkeev property in spaces of continuous functions (II)

We prove that for each Polish space X, the space C(X) of continuous real-valued functions on X satisfies a strong version of the Pytkeev property, if endowed with the compact-open topology. (This shows that whereas it need not be metrizable, it is "very close" to that.) We also consider the Pytkeev property in the case where C(X) is endowed with the topology of pointwise convergence.

math.GN↗