In this paper, we present an efficient attack to the multivariate Quadratic Quasigroups (MQQ) cryptosystem. Our cryptanalysis breaks MQQ cryptosystems by solving systems of multivariate quadratic polynomial equations using a modified version of the MutantXL algorithm. We present experimental results comparing the behavior of our implementation of MutantXL to Magma's implementation of F_4 on MQQ systems (\ge 135 bit). Based on our results we show that the MutantXL implementation solves with much less memory than Magma's implementation of F_4 algorithm.

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The concept of matrix canonical realization of a Lie algebra is introduced. The generators of the Lie algebra of the pseudoorthogonal group n) are recurrently expressed in terms of matrices with polyno-mial elements in a certain number of quantum-mechanical canonical variables p~, qi and they depend on a certain number of free real para-meters. The realizations are, in the well-defined sense, skew-hermitean and Casimir operators are multiples of the identity element. Part of them are usual canonical realizations.

We consider the Laplacian in a straight planar strip with Dirichlet boundary which has two Neumann'windows' of the same length, the centres of which are 2l apart, and study the asymptotic behaviour of the discrete spectrum as l. It is shown that there are pairs of eigenvalues around each isolated eigenvalue of a single-window strip and their distances vanish exponentially in the limit l. We derive an asymptotic expansion also in the case where a single window gives rise to a threshold resonance which the presence of the other window turns into a single isolated eigenvalue.