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 $X$ satisfies the Hurewicz property if, and only if, each large open cover of $X$ 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 The attack works for any group.Comment: This paper has been withdrawn by the author.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