Trading group theory for randomness
STOC '85 Proceedings of the seventeenth annual ACM symposium on Theory of computing
STOC '87 Proceedings of the nineteenth annual ACM symposium on Theory of computing
Arthur-Merlin games: a randomized proof system, and a hierarchy of complexity class
Journal of Computer and System Sciences - 17th Annual ACM Symposium in the Theory of Computing, May 6-8, 1985
Journal of the ACM (JACM)
Bounded round interactive proofs in finite groups
SIAM Journal on Discrete Mathematics
Journal of Computer and System Sciences
Hypergraph isomorphism and structural equivalence of Boolean functions
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
On the nlog n isomorphism technique (A Preliminary Report)
STOC '78 Proceedings of the tenth annual ACM symposium on Theory of computing
Computation with permutation groups
SYMSAC '71 Proceedings of the second ACM symposium on Symbolic and algebraic manipulation
STOC '83 Proceedings of the fifteenth annual ACM symposium on Theory of computing
Linear time algorithms for Abelian group isomorphism and related problems
Journal of Computer and System Sciences
Polynomial-time algorithms for permutation groups
SFCS '80 Proceedings of the 21st Annual Symposium on Foundations of Computer Science
Computational complexity and the classification of finite simple groups
SFCS '83 Proceedings of the 24th Annual Symposium on Foundations of Computer Science
Polynomial-time theory of matrix groups
Proceedings of the forty-first annual ACM symposium on Theory of computing
ICALP '09 Proceedings of the 36th International Colloquium on Automata, Languages and Programming: Part I
Is code equivalence easy to decide?
IEEE Transactions on Information Theory
Polynomial-time isomorphism test for groups with no abelian normal subgroups
ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part I
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The isomorphism problem for groups given by their multiplication tables has long been known to be solvable in time nlog n+O(1). The decades-old quest for a polynomial-time algorithm has focused on the very difficult case of class-2 nilpotent groups (groups whose quotient by their center is abelian), with little success. In this paper we consider the opposite end of the spectrum and initiate a more hopeful program to find a polynomial-time algorithm for semisimple groups, defined as groups without abelian normal subgroups. First we prove that the isomorphism problem for this class can be solved in time nO(log log n). We then identify certain bottlenecks to polynomial-time solvability and give a polynomial-time solution to a rich subclass, namely the semisimple groups where each minimal normal subgroup has a bounded number of simple factors. We relate the results to the filtration of groups introduced by Babai and Beals (1999). One of our tools is an algorithm for equivalence of (not necessarily linear) codes in simply-exponential time in the length of the code, obtained by modifying Luks's algorithm for hypergraph isomorphism in simply-exponential time in the number of vertices (FOCS 1999). We comment on the complexity of the closely related problem of permutational isomorphism of permutation groups.