Graph Isomorphism is Not AC0-Reducible to Group Isomorphism

We give a new upper bound for the Group and Quasigroup Isomorphism problems when the input structures are given explicitly by multiplication tables. We show that these problems can be computed by polynomial size nondeterministic circuits of unbounded fan-in with O (log log n ) depth and O (log 2 n ) nondeterministic bits, where n is the number of group elements. This improves the existing upper bound for the problems. In the previous bound the circuits have bounded fan-in but depth O (log 2 n ) and also O (log 2 n ) nondeterministic bits. We then prove that the kind of circuits from our upper bound cannot compute the Parity function. Since Parity is AC 0 -reducible to Graph Isomorphism, this implies that Graph Isomorphism is strictly harder than Group or Quasigroup Isomorphism under the ordering defined by AC 0 reductions. We extend this result to the stronger ACC 0 [ p ] reduction and its randomized version.