Embedding-setup problem

Description
Embedding-setup problems are problems where the groups involved are described as subgroups of some big universe group such as the general linear group or the symmetric group (group of all permutations).

The universe group
The universe group is some big group where:


 * Every element of the universe group has an encoding that is roughly of size equalling the logarithm of the size of the universe group.
 * Multiplication of elements in the universe group (in terms of their encodings) takes tiem polynomial in the code lengths of the elements.
 * The inverse can also be computed (in terms of the encodings) in time polynomial in the code lengths of the elements.

The universe group can thus be treated as a black-box group where the multiplication and inversion algorithms take polynomial time.

Of course, in reality, the universe group would be the symmetric group or the general linear group, something which has far more structure than a black-box group. However, the point is that any black-box algorithm will work on any universe group.

The group description
The group is described by means of a generating set, where each element of the generating set is given by means of its code in the universe group.

Alternatively, the group can be described by means of a membership test for it.

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