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106
Succinct Quantum Proofs for Properties of Finite Groups
 In Proc. IEEE FOCS
, 2000
"... In this paper we consider a quantum computational variant of nondeterminism based on the notion of a quantum proof, which is a quantum state that plays a role similar to a certificate in an NPtype proof. Specifically, we consider quantum proofs for properties of blackbox groups, which are finite g ..."
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Cited by 86 (3 self)
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In this paper we consider a quantum computational variant of nondeterminism based on the notion of a quantum proof, which is a quantum state that plays a role similar to a certificate in an NPtype proof. Specifically, we consider quantum proofs for properties of blackbox groups, which are finite groups whose elements are encoded as strings of a given length and whose group operations are performed by a group oracle. We prove that for an arbitrary group oracle there exist succinct (polynomiallength) quantum proofs for the Group NonMembership problem that can be checked with small error in polynomial time on a quantum computer. Classically this is impossibleit is proved that there exists a group oracle relative to which this problem does not have succinct proofs that can be checked classically with bounded error in polynomial time (i.e., the problem is not in MA relative to the group oracle constructed). By considering a certain subproblem of the Group NonMembership problem we obtain a simple proof that there exists an oracle relative to which BQP is not contained in MA. Finally, we show that quantum proofs for nonmembership and classical proofs for various other group properties can be combined to yield succinct quantum proofs for other group properties not having succinct proofs in the classical setting, such as verifying that a number divides the order of a group and verifying that a group is not a simple group.
Generating random elements of a finite group
 Comm. Algebra
, 1995
"... We present a “practical ” algorithm to construct random elements of a finite group. We analyse its theoretical behaviour and prove that asymptotically it produces uniformly distributed tuples of elements. We discuss tests to assess its effectiveness and use these to decide when its results are accep ..."
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Cited by 83 (13 self)
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We present a “practical ” algorithm to construct random elements of a finite group. We analyse its theoretical behaviour and prove that asymptotically it produces uniformly distributed tuples of elements. We discuss tests to assess its effectiveness and use these to decide when its results are acceptable for some matrix groups. 1 1
Selftesting/correcting for polynomials and for approximate functions
 in Proceedings of the 23rd Annual Symposium on Theory of Computing (STOC
, 1991
"... The study of selftesting/correcting programs was introduced in [8] in order to allow one to use program P to compute function f without trusting that P works correctly. A selftester for f estimates the fraction of x for which P (x) = f(x); and a selfcorrector for f takes a program that is correc ..."
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Cited by 81 (15 self)
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The study of selftesting/correcting programs was introduced in [8] in order to allow one to use program P to compute function f without trusting that P works correctly. A selftester for f estimates the fraction of x for which P (x) = f(x); and a selfcorrector for f takes a program that is correct on most inputs and turns it into a program that is correct on every input with high probability 1. Both access P only as a blackbox and in some precise way are not allowed to compute the function f. Selfcorrecting is usually easy when the function has the random selfreducibility property. One class of such functions that has this property is the class of multivariate polynomials over finite fields [4] [12]. We extend this result in two directions. First, we show that polynomials are random selfreducible over more general domains: specifically, over the rationals and over noncommutative rings. Second, we show that one can get selfcorrectors even when the program satisfies weaker conditions, i.e. when the program has more errors, or when the program behaves in a more adversarial manner by changing the function it computes between successive calls. Selftesting is a much harder task. Previously it was known how to selftest for a few special examples of functions, such as the class of linear functions. We show that one can selftest the whole class of polynomial functions over Zp for prime p.
The product replacement algorithm and Kazhdan’s property
 T), J. Amer. Math. Soc
"... A problem of great importance in computational group theory is to generate (nearly) uniformly distributed random elements in a finite group G. A good example of such an algorithm should start at any given set of generators, use no prior knowledge of the structure of G, and in a polynomial number of ..."
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Cited by 53 (13 self)
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A problem of great importance in computational group theory is to generate (nearly) uniformly distributed random elements in a finite group G. A good example of such an algorithm should start at any given set of generators, use no prior knowledge of the structure of G, and in a polynomial number of group operations
A polynomialtime theory of blackbox groups I
, 1998
"... We consider the asymptotic complexity of algorithms to manipulate matrix groups over finite fields. Groups are given by a list of generators. Some of the rudimentary tasks such as membership testing and computing the order are not expected to admit polynomialtime solutions due to number theoretic o ..."
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Cited by 47 (6 self)
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We consider the asymptotic complexity of algorithms to manipulate matrix groups over finite fields. Groups are given by a list of generators. Some of the rudimentary tasks such as membership testing and computing the order are not expected to admit polynomialtime solutions due to number theoretic obstacles such as factoring integers and discrete logarithm. While these and other “abelian obstacles ” persist, we demonstrate that the “nonabelian normal structure ” of matrix groups over finite fields can be mapped out in great detail by polynomialtime randomized (Monte Carlo) algorithms. The methods are based on statistical results on finite simple groups. We indicate the elements of a project under way towards a more complete “recognition” of such groups in polynomial time. In particular, under a now plausible hypothesis, we are able to determine the names of all nonabelian composition factors of a matrix group over a finite field. Our context is actually far more general than matrix groups: most of the algorithms work for “blackbox groups ” under minimal assumptions. In a blackbox group, the group elements are encoded by strings of uniform length, and the group operations are performed by a “black box.”
What Do We Know About The Product Replacement Algorithm?
 in: Groups ann Computation III
, 2000
"... . The product replacement algorithm is a commonly used heuristic to generate random group elements in a finite group G, by running a random walk on generating ktuples of G. While experiments showed outstanding performance, until recently there was little theoretical explanation. We give an exten ..."
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Cited by 47 (8 self)
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. The product replacement algorithm is a commonly used heuristic to generate random group elements in a finite group G, by running a random walk on generating ktuples of G. While experiments showed outstanding performance, until recently there was little theoretical explanation. We give an extensive review of both positive and negative theoretical results in the analysis of the algorithm. Introduction In the past few decades the study of groups by means of computations has become a wonderful success story. The whole new field, Computational Group Theory, was developed out of needs to discover and prove new results on finite groups. More recently, the probabilistic method became an important tool for creating faster and better algorithms. A number of applications were developed which assume a fast access to (nearly) uniform group elements. This led to a development of the so called "product replacement algorithm", which is a commonly used heuristic to generate random group elemen...
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 45 (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.