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Quantum algorithms for solvable groups
 In Proceedings of the 33rd ACM Symposium on Theory of Computing
, 2001
"... ABSTRACT In this paper we give a polynomialtime quantum algorithm for computing orders of solvable groups. Several other problems, such as testing membership in solvable groups, testing equality of subgroups in a given solvable group, and testing normality of a subgroup in a given solvable group, r ..."
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Cited by 37 (1 self)
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ABSTRACT In this paper we give a polynomialtime quantum algorithm for computing orders of solvable groups. Several other problems, such as testing membership in solvable groups, testing equality of subgroups in a given solvable group, and testing normality of a subgroup in a given solvable group, reduce to computing orders of solvable groups and therefore admit polynomialtime quantum algorithms as well. Our algorithm works in the setting of blackbox groups, wherein none of these problems have polynomialtime classical algorithms. As an important byproduct, our algorithm is able to produce a pure quantum state that is uniform over the elements in any chosen subgroup of a solvable group, which yields a natural way to apply existing quantum algorithms to factor groups of solvable groups. 1.
Separation of Variables and the Computation of Fourier Transforms on Finite Groups, I
 I. J. OF THE AMER. MATH. SOC
, 1997
"... This paper introduces new techniques for the efficient computation of a Fourier transform on a finite group. We present a divide and conquer approach to the computation. The divide aspect uses factorizations of group elements to reduce the matrix sum of products for the Fourier transform to simpler ..."
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Cited by 17 (7 self)
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This paper introduces new techniques for the efficient computation of a Fourier transform on a finite group. We present a divide and conquer approach to the computation. The divide aspect uses factorizations of group elements to reduce the matrix sum of products for the Fourier transform to simpler sums of products. This is the separation of variables algorithm. The conquer aspect is the final computation of matrix products which we perform efficiently using a special form of the matrices. This form arises from the use of subgroupadapted representations and their structure when evaluated at elements which lie in the centralizers of subgroups in a subgroup chain. We present a detailed analysis of the matrix multiplications arising in the calculation and obtain easytouse upper bounds for the complexity of our algorithm in terms of representation theoretic data for the group of interest. Our algorithm encompasses many of the known examples of fast Fourier transforms. We recover the b...
Simple groups in computational group theory
 Proceedings of the International Congress of Mathematicians
, 1998
"... ..."
THE HIDDEN SUBGROUP PROBLEM  REVIEW AND OPEN PROBLEMS
, 2004
"... An overview of quantum computing and in particular the Hidden Subgroup Problem are presented from a mathematical viewpoint. Detailed proofs are supplied for many important results from the literature, and notation is unified, making it easier to absorb the background necessary to begin research on ..."
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Cited by 12 (1 self)
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An overview of quantum computing and in particular the Hidden Subgroup Problem are presented from a mathematical viewpoint. Detailed proofs are supplied for many important results from the literature, and notation is unified, making it easier to absorb the background necessary to begin research on the Hidden Subgroup Problem. Proofs are provided which give very concrete algorithms and bounds for the finite abelian case with little outside references, and future directions are provided for the nonabelian case. This summary is current as of October 2004.
Algorithms for Matrix Groups and the Tits Alternative
 Proc. 36th IEEE FOCS
, 1999
"... l over the generators grows as c l for some constant c>1 depending on G. For groups with abelian subgroups of finite index, we obtain a Las Vegas algorithm for several basic computational tasks, including membership testing and computing a presentation. This generalizes recent work of Beals ..."
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Cited by 11 (2 self)
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l over the generators grows as c l for some constant c>1 depending on G. For groups with abelian subgroups of finite index, we obtain a Las Vegas algorithm for several basic computational tasks, including membership testing and computing a presentation. This generalizes recent work of Beals and Babai, who give a Las Vegas algorithm for the case of finite groups, as well as recent work of Babai, Beals, Cai, Ivanyos, and Luks, who give a deterministic algorithm for the case of abelian groups. # 1999 Academic Press Article ID jcss.1998.1614, available online at http:##www.idealibrary.com on 260 00220000#99 #30.00 Copyright # 1999 by Academic Press All rights of reproduction in any form reserved. * Research conducted while visiting IAS and DIMACS and supported in part by an NSF Mathematical Sciences
Solvable BlackBox Group Problems are Low for PP
 Theoretical Computer Science
, 1997
"... Let B = fBmgm>0 be a countable family of nite groups whose elements are uniquely encoded as strings of uniform length, and group operations are computable in time bounded by a polynomial in m. A blackbox group over a group family B is a subgroup of some member of B and is presented by a gener ..."
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Cited by 10 (3 self)
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Let B = fBmgm>0 be a countable family of nite groups whose elements are uniquely encoded as strings of uniform length, and group operations are computable in time bounded by a polynomial in m. A blackbox group over a group family B is a subgroup of some member of B and is presented by a generator set. In this paper we study the complexity of several algorithmic problems for abelian blackbox groups and solvable blackbox groups. We design a suitable oracle algorithm that computes an independent set of generators for a given abelian blackbox group. Using this we show that the problems of Membership Testing, Group Intersection, Order Verication, and Group Isomorphism over abelian blackbox groups are in SPP. We also show that Group Factorization, Coset Intersection, and Double Coset Membership problems over abelian blackbox groups are in LWPP. As a consequence, all these problems are low for PP and C=P. We dene the notion of canonical generator sets for classes of groups ...
Constructions in publickey cryptography over matrix groups
 Contemp. Math., Amer. Math. Soc
"... The purpose of the paper is to give new key agreement protocols (a multiparty extension of the protocol due to AnshelAnshelGoldfeld and a generalization of the DiffieHellman protocol from abelian to solvable groups) and a new homomorphic publickey cryptosystem. They rely on difficulty of the co ..."
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Cited by 9 (6 self)
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The purpose of the paper is to give new key agreement protocols (a multiparty extension of the protocol due to AnshelAnshelGoldfeld and a generalization of the DiffieHellman protocol from abelian to solvable groups) and a new homomorphic publickey cryptosystem. They rely on difficulty of the conjugacy and membership problems for subgroups of a given group. To support these and other known cryptographic schemes we present a general technique to produce a family of instances being matrix groups (over finite commutative rings) which play a role for these schemes similar to the groups Z ∗ n in the existing cryptographic constructions like RSA or discrete logarithm. Partially supported by RFFI, grants, 030100349, NSH2251.2003.1. The paper was done during the