  We study the computational power of deciding whether a given truth-table can
be described by a circuit of a given size (the Minimum Circuit Size Problem, or
MCSP for short), and of the variant denoted as MKTP where circuit size is
replaced by a polynomially-related Kolmogorov measure. All prior reductions
from supposedly-intractable problems to MCSP / MKTP hinged on the power of MCSP
/ MKTP to distinguish random distributions from distributions produced by
hardness-based pseudorandom generator constructions. We develop a fundamentally
different approach inspired by the well-known interactive proof system for the
complement of Graph Isomorphism (GI). It yields a randomized reduction with
zero-sided error from GI to MKTP. We generalize the result and show that GI can
be replaced by any isomorphism problem for which the underlying group satisfies
some elementary properties. Instantiations include Linear Code Equivalence,
Permutation Group Conjugacy, and Matrix Subspace Conjugacy. Along the way we
develop encodings of isomorphism classes that are efficiently decodable and
achieve compression that is at or near the information-theoretic optimum; those
encodings may be of independent interest.
