Beating the Generator-Enumeration Bound for pp-Group Isomorphism

We consider the group isomorphism problem: given two finite groups G and H specified by their multiplication tables, decide if G cong H. For several decades, the n^(log_p n + O(1)) generator-enumeration bound (where p is the smallest prime dividing the order of the group) has been the best worst-case result for general groups. In this work, we show the first improvement over the generator-enumeration bound for p-groups, which are believed to be the hard case of the group isomorphism problem. We start by giving a Turing reduction from group isomorphism to n^((1 / 2) log_p n + O(1)) instances of p-group composition-series isomorphism. By showing a Karp reduction from p-group composition-series isomorphism to testing isomorphism of graphs of degree at most p + O(1) and applying algorithms for testing isomorphism of graphs of bounded degree, we obtain an n^(O(p)) time algorithm for p-group composition-series isomorphism. Combining these two results yields an algorithm for p-group isomorphism that takes at most n^((1 / 2) log_p n + O(p)) time. This algorithm is faster than generator-enumeration when p is small and slower when p is large. Choosing the faster algorithm based on p and n yields an upper bound of n^((1 / 2 + o(1)) log n) for p-group isomorphism.Comment: 15 pages. This is an updated and improved version of the results for p-groups in arXiv:1205.0642 and TR11-052 in ECC

An Efficient Quantum Algorithm for some Instances of the Group Isomorphism Problem

In this paper we consider the problem of testing whether two finite groups are isomorphic. Whereas the case where both groups are abelian is well understood and can be solved efficiently, very little is known about the complexity of isomorphism testing for nonabelian groups. Le Gall has constructed an efficient classical algorithm for a class of groups corresponding to one of the most natural ways of constructing nonabelian groups from abelian groups: the groups that are extensions of an abelian group $A$ by a cyclic group $Z_m$ with the order of $A$ coprime with $m$. More precisely, the running time of that algorithm is almost linear in the order of the input groups. In this paper we present a quantum algorithm solving the same problem in time polynomial in the logarithm of the order of the input groups. This algorithm works in the black-box setting and is the first quantum algorithm solving instances of the nonabelian group isomorphism problem exponentially faster than the best known classical algorithms.Comment: 20 pages; this is the full version of a paper that will appear in the Proceedings of the 27th International Symposium on Theoretical Aspects of Computer Science (STACS 2010

An Efficient Quantum Algorithm for Some Instances of the Group Isomorphism Problem

In this paper we consider the problem of testing whether two finite groups are isomorphic. Whereas the case where both groups are abelian is well understood and can be solved efficiently, very little is known about the complexity of isomorphism testing for nonabelian groups. Le Gall has constructed an efficient classical algorithm for a class of groups corresponding to one of the most natural ways of constructing nonabelian groups from abelian groups: the groups that are extensions of an abelian group $A$ by a cyclic group $Int_m$ with the order of $A$ coprime with $m$. More precisely, the running time of that algorithm is almost linear in the order of the input groups. In this paper we present a emph{quantum} algorithm solving the same problem in time polynomial in the emph{logarithm} of the order of the input groups. This algorithm works in the black-box setting and is the first quantum algorithm solving instances of the nonabelian group isomorphism problem exponentially faster than the best known classical algorithms