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For Discrete Logarithm based cryptography, it is desirable to find efficiently implementable groups for which sub-exponential algorithms for the Discrete Logarithm Problem are not available. Thus far, the only candidates for such groups seem to be (carefully chosen) groups of points on elliptic curves. Groups of invertible matrices over a finite field, where proved by Menezes and Wu inadequate for this purpose. Consequently, any candidate for a platform group for Discrete Logarithm based cryptography must not be efficiently embeddable in a group of matrices.
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