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We show that Graph Isomorphism is in the complexity class SPP, and hence it is in ⊕P (in fact, it is in Mod k P for each k ≥ 2). We derive this result as a corollary of a more general result: we show that a generic problem FIND-GROUP has an FP SPP algorithm. This general result has other consequences: for example, it follows that the hidden subgroup problem… (More)

It is shown that the assumption of NP having polynomial-size circuits implies (apart from a collapse of the polynomial-time hierarchy as shown by Karp and Lip-ton) that the classes AM and MA of Babai's Arthur-Merlin hierarchy coincide. This means that also a certain inner collapse of the remaining classes of the polynomial-time hierarchy occurs. It is well… (More)

In this paper we study the complexity of Bounded Color Multiplic-ity Graph Isomorphism BCGI b : the input is a pair of vertex-colored graphs such that the number of vertices of a given color in an input graph is bounded by b. We show that BCGI b is in the #L hierarchy (more precisely, the Mod k L hierarchy for some constant k depending on b). Combined with… (More)

This paper is motivated by the computer-generated nonadditive code described in Rains et al [RHSN97]. We describe a theory of non-stabilizer codes of which the nonadditive code of Rains et al is an example. Furthermore, we give a general strategy of constructing good nonstabilizer codes from good stabilizer codes and give some explicit constructions and… (More)

We study the complexity of some computational problems on finite black-box rings whose elements are encoded as strings of a given length and the ring operations are performed by a black-box oracle. We give a polynomial-time quantum algorithm to compute a basis representation for a given black-box ring. Using this result we obtain polynomial-time quantum… (More)

In this paper we study the complexity of sets that reduce to sparse sets (and tally sets), and the complexity of the simplest sparse sets to which such sets reduce. We show even with respect to very exible reductions that NP cannot have sparse hard sets unless P = NP; an immediate consequence of our results is: If any NP-complete set conjunctively reduces… (More)

In this paper we extend a key result of Nisan and Wigderson NW94] to the nondeterministic setting: for all > 0 we show that if there is a language in E = DTIME(2 O(n)) that is hard to approximate by nondeterministic circuits of size 2 n , then there is a pseudorandom generator that can be used to derandomize BP NP (in symbols, BP NP = NP). By applying this… (More)