The complexity of promise problems with applications to public-key cryptography
Information and Control
Trading group theory for randomness
STOC '85 Proceedings of the seventeenth annual ACM symposium on Theory of computing
Graph isomorphism is in the low hierarchy
4th Annual Symposium on Theoretical Aspects of Computer Sciences on STACS 87
Does co-NP have short interactive proofs?
Information Processing Letters
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
Structural complexity 2
Bounded round interactive proofs in finite groups
SIAM Journal on Discrete Mathematics
Turing machines with few accepting computations and low sets for PP
Journal of Computer and System Sciences
The graph isomorphism problem: its structural complexity
The graph isomorphism problem: its structural complexity
A taxonomy of complexity classes of functions
Journal of Computer and System Sciences
Computing functions with parallel queries to NP
Theoretical Computer Science
Solvable black-box group problems are low for PP
Theoretical Computer Science
Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
SIAM Journal on Computing
Normal subgroup reconstruction and quantum computation using group representations
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
Efficient quantum algorithms for some instances of the non-Abelian hidden subgroup problem
Proceedings of the thirteenth annual ACM symposium on Parallel algorithms and architectures
Complexity Limitations on Quantum Computation
COCO '98 Proceedings of the Thirteenth Annual IEEE Conference on Computational Complexity
STOC '83 Proceedings of the fifteenth annual ACM symposium on Theory of computing
Counting Complexity of Solvable Black-Box Group Problems
SIAM Journal on Computing
Computational complexity of computing a partial solution for the Graph Automorphism problems
Theoretical Computer Science
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We show that Graph Isomorphism is in the complexity class SPP, and hence it is in ⊕P (in fact, in ModkP for each k ≥ 2). These inclusions for Graph Isomorphism were not known prior to membership in SPP. We derive this result as a corollary of a more general result: we show that a generic problem FIND-GROUP has an FPSPP algorithm. This general result has other consequences: for example, it follows that the hidden subgroup problem for permutation groups, studied in the context of quantum algorithms, has an FPSPP algorithm. Also, some other algorithmic problems over permutation groups known to be at least as hard as Graph Isomorphism (e.g., coset intersection) are in SPP, and thus in ModkP for each k ≥ 2.