202 research outputs found
The Hurewicz covering property and slaloms in the Baire space
According to a result of Kocinac and Scheepers, the Hurewicz covering
property is equivalent to a somewhat simpler selection property: For each
sequence of large open covers of the space one can choose finitely many
elements from each cover to obtain a groupable cover of the space. We simplify
the characterization further by omitting the need to consider sequences of
covers: A set of reals satisfies the Hurewicz property if, and only if,
each large open cover of contains a groupable subcover. This solves in the
affirmative a problem of Scheepers.
The proof uses a rigorously justified abuse of notation and a "structure"
counterpart of a combinatorial characterization, in terms of slaloms, of the
minimal cardinality b of an unbounded family of functions in the Baire space.
In particular, we obtain a new characterization of \b.Comment: Small update
On an authentication scheme based on the Root Problem in the braid group
Lal and Chaturvedi proposed two authentication schemes based on the
difficulty of the Root Problem in the braid group. We point out that the first
scheme is not really as secure as the Root Problem, and describe an efficient
way to crack it. The attack works for any group.Comment: This paper has been withdrawn by the author. One of the claims is
incorrect as written. We are working on correcting and generalizing it. This
will be published in another pape
Fast generators for the Diffie-Hellman key agreement protocol and malicious standards
The Diffie-Hellman key agreement protocol is based on taking large powers of
a generator of a prime-order cyclic group. Some generators allow faster
exponentiation. We show that to a large extent, using the fast generators is as
secure as using a randomly chosen generator. On the other hand, we show that if
there is some case in which fast generators are less secure, then this could be
used by a malicious authority to generate a standard for the Diffie-Hellman key
agreement protocol which has a hidden trapdoor.Comment: Small update
Selection principles in mathematics: A milestone of open problems
We survey some of the major open problems involving selection principles,
diagonalizations, and covering properties in topology and infinite
combinatorics. Background details, definitions and motivations are also
provided.Comment: Small update
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