- Authors: E. Allender, J. Grochow, D. van Melkebeek, C. Moore, and A. Morgan

- Status: To appear in
*SIAM Journal on Computing*.

- Earlier versions:
- E. Allender, J. Grochow, D. van Melkebeek, C. Moore, and A. Morgan.
Minimum Circuit Size, Graph Isomorphism,
and Related Problems.
In
*Proceedings of the 9th Innovations in Theoretical Computer Science Conference*,
pages 20:1-20, 2018.
- E. Allender, J. Grochow, D. van Melkebeek, C. Moore, and A. Morgan.
Minimum Circuit Size, Graph Isomorphism,
and Related Problems.
Electronic Colloquium on Computational Complexity, Technical Report ECCC-TR 17-158, 2017.
- E. Allender, J. Grochow, D. van Melkebeek, C. Moore, and A. Morgan.
A note on Graph Automorphism and Smart Reductions.
Electronic Colloquium on Computational Complexity, Technical Report ECCC-TR 15-162, 2015.

#### Abstract

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 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.