We provide new provable polynomial time solutions
of a number of problems in noncommutative-algebraic cryptography.
In contrast to the linear centralizer method of \cite{LinCent}, the new method is
very simple: In order to solve linear equations on matrices
restricted to matrix groups, solve them over the generated
algebras. We name this approach the \emph{algebraic span method}.
The resulting algorithms are have substantially better performance than those of \cite{LinCent}.
These algorithms constitute cryptanalyses of the motivating protocols that
cannot be foiled by changing the distributions used in the protocols, and are
practical for most affordable parameter settings.
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