In this paper we analyze the Kahrobaei-Lam-Shpilrain (KLS) key exchange protocols that use extensions by endomorpisms of matrices over a Galois field proposed in \cite{Kahrobaei-Lam-Shpilrain:2014}. We show that both protocols are vulnerable to a simple linear algebra attack.

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We establish the full global non-linear stability of the Kerr–de Sitter family of black holes, as solutions of the initial value problem for the Einstein vacuum equations with positive cosmological constant, for small angular momenta, and without any symmetry assumptions on the initial data. We achieve this by extending the linear and non-linear analysis on black hole spacetimes described in a sequence of earlier papers by the authors: we develop a general framework which enables us to deal systematically with the diffeomorphism invariance of Einstein’s equations. In particular, the iteration scheme used to solve Einstein’s equations automatically finds the parameters of the Kerr–de Sitter black hole that the solution is asymptotic to, the exponentially decaying tail of the solution, and the gauge in which we are able to find the solution; the gauge here is a wave map/DeTurck type gauge, modified by source terms which are treated as unknowns, lying in a suitable finite-dimensional space.

We give complete detail of the description of the GNS representation of the quantum plane A and its dual  as a von Neumann algebra. In particular, we obtain a rather surprising result that the multiplicative unitary W is manageable in this quantum semigroup context. We study the quantum double group construction introduced by Woronowicz, and using Baaj and Vaes' construction of the multiplicative unitary Wm, we give the GNS description of the quantum double D(A) which is equivalent to GLq+(2, ℝ). Furthermore, we study the fundamental corepresentation Tλ, t and its matrix coefficients, and show that it can be expressed by the b-hypergeometric function. We also study the regular corepresentation and representation induced by Wm and prove that the space of L2 functions on the quantum double decomposes into the continuous series representation of Uqq̃(gl(2, ℝ)) with the quantum dilogarithm {pipe}Sb(Q+2iα){pipe}2 as the Plancherel measure. Finally, we describe certain representation theoretic meaning of integral transforms involving the quantum dilogarithm function.