We introduce the
linear centralizer method
, and use it to devise a provable polynomial-time solution of the Commutator Key Exchange Problem, the computational problem on which, in the passive ...adversary model, the security of the Anshel–Anshel–Goldfeld (Anshel et al., Math. Res. Lett. 6:287–291,
1999
)
Commutator
key exchange protocol is based. We also apply this method to solve, in polynomial time, the computational problem underlying the
Centralizer
key exchange protocol, introduced by Shpilrain and Ushakov in (Contemp. Math. 418:161–167,
2006
).
This is the first provable polynomial-time cryptanalysis of the Commutator key exchange protocol, hitherto the most important key exchange protocol in the realm of noncommutative algebraic cryptography, and the first cryptanalysis (of any kind) of the Centralizer key exchange protocol. Unlike earlier cryptanalyses of the Commutator key exchange protocol, our cryptanalyses cannot be foiled by changing the distributions used in the protocol.
We construct Menger subsets of the real line whose product is not Menger in the plane. In contrast to earlier constructions, our approach is purely combinatorial. The set theoretic hypothesis used in ...our construction is far milder than earlier ones, and holds in almost all canonical models of set theory of the real line. On the other hand, we establish productive properties for versions of Menger's property parameterized by filters and semifilters. In particular, the Continuum Hypothesis implies that every productively Menger set of real numbers is productively Hurewicz, and each ultrafilter version of Menger's property is strictly between Menger's and Hurewicz's classic properties. We include a number of open problems emerging from this study.
We study productive properties of \gamma Solving a problem of F. Jordan, we show that for every unbounded tower set X\subseteq \mathbb{R}, the space \operatorname {C}_\mathrm {p}(X) is productively ...\gamma Solving problems of Scheepers and Weiss and proving a conjecture of Babinkostova-Scheepers, we prove that, assuming the Continuum Hypothesis, there are \gamma Solving a problem of Scheepers-Tall, we show that the properties \gamma We apply our results to solve a large number of additional problems and use Arhangel'skiĭ duality to obtain results concerning local properties of function spaces and countable topological groups.
I provide simplified proofs for each of the following fundamental theorems regarding selection principles:
(1)
The Quasinormal Convergence Theorem, due to the author and Zdomskyy, asserting that a ...certain, important property of the space of continuous functions on a space is actually preserved by Borel images of that space.
(2)
The Scheepers Diagram Last Theorem, due to Peng, completing all provable implications in the diagram.
(3)
The Menger Game Theorem, due to Telgársky, determining when Bob has a winning strategy in the game version of Menger’s covering property.
(4)
A lower bound on the additivity of Rothberger’s covering property, due to Carlson.
The simplified proofs lead to several new results.
The Haar measure problem Przeździecki, Adam J.; Szewczak, Piotr; Tsaban, Boaz
Proceedings of the American Mathematical Society,
03/2019, Letnik:
147, Številka:
3
Journal Article
Recenzirano
An old problem asks whether every compact group has a Haar-nonmeasurable subgroup. A series of earlier results reduced the problem to infinite metrizable profinite groups. We provide a positive ...answer, assuming a weak, potentially provable, consequence of the Continuum Hypothesis. We also establish the dual, Baire category analogue of this result.
We settle all problems concerning the additivity of the Gerlits-Nagy property and related additivity numbers posed by Scheepers in his tribute paper to Gerlits. 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. Our applications extend earlier results of Babinkostova and Scheepers.
The fully homomorphic symmetric encryption scheme MORE encrypts
random keys by conjugation with a random invertible matrix over an RSA modulus.
We provide a known-ciphertext cryptanalysis recovering ...a linear dependence
among any pair of encrypted keys.
We provide conceptual proofs of the two most fundamental theorems concerning topological games and open covers: Hurewicz's Theorem concerning the Menger game, and Pawlikowski's Theorem concerning the ...Rothberger game.
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