Small generating set-finding problem for black-box groups

Description
We are given a group $$G$$ of order $$N$$ is specified by means of an encoding. We need to find a generating set for $$G$$ of size at most $$\log_2N$$.

A somewhat stronger requirement (which will guarantee this one) is: we need to find a generating set in which no element is redundant (that is, throwing out any element makes the subset non-generating).

Nondeterministic algorithms
The generating set-finding problem has a clear nondeterministic reduction to the generating set-checking problem -- first nondeterministically obtain a generating set, then invoke the generating set-checking problem to verify that it is genuinely a generating set.