Polynomial time algorithms for modules over finite dimensional algebras

Chistov, A. L.; Ivanyos, Gábor; Karpiński, Marek

doi:10.1145/258726.258751

Cited by 50 publications

(89 citation statements)

References 14 publications

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“…One of the problems solved in [16] is to decide if W = V 1 ∩ ... ∩ V K contains a nonsingular matrix. It is not clear to the author whether the class of such linear subspaces W satisfies the (ERP) property.…”

Section: Suppose That Linear Subspacesmentioning

confidence: 99%

Classical deterministic complexity of Edmonds' Problem and quantum entanglement

Gurvits

2003

Proceedings of the Thirty-Fifth Annual ACM Symposium on Theory of Computing

390

354

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Generalizing a decision problem for bipartite perfect matching , J. Edmonds introduced in [15] the problem (now known as the Edmonds Problem) of deciding if a given linear subspace of M (N ) contains a nonsingular matrix, where M (N ) stands for the linear space of complex N × N matrices. This problem led to many fundamental developments in matroid theory etc. Classical matching theory can be defined in terms of matrices with nonnegative entries. The notion of Positive operator, central in Quantum Theory, is a natural generalization of matrices with nonnegative entries. (Here operator refers to maps from matrices to matrices.) First, we reformulate the Edmonds Problem in terms of of completely positive operators, or equivalently, in terms of bipartite density matrices . It turns out that one of the most important cases when Edmonds' problem can be solved in polynomial deterministic time, i.e. an intersection of two geometric matroids, corresponds to unentangled (aka separable ) bipartite density matrices . We introduce a very general class (or promise ) of linear subspaces of M (N ) on which there exists a polynomial deterministic time algorithm to solve Edmonds' problem . The algorithm is a thoroughgoing generalization of algorithms in [29], [38], and its analysis benefits from an operator analog of permanents, so called Quantum Permanents . Finally, we prove that the weak membership problem for the convex set of separable normalized bipartite density matrices is NP-HARD. Introduction and Main DefinitionsLet M (N ) be the linear space of N × N complex matrices . The following fundamental problem has been posed by J. Edmonds in [15]: Problem 1.1: Given a linear subspace V ⊂ M (N ) to decide if there exists a nonsingular matrix A ∈ V .We will assume throughout the paper that the subspace V is presented as a finite spanning k-tuple of rational matrices S(V ) = {A 1 , ..., A k }(k ≤ N 2 ), i.e . the linear space generated by them is equal to V . As usual, the complexity parameter of the input < S(V ) > is equal to ( N + "number of bits of entries of matrices A i , 1 ≤ i ≤ k" ). Thus Edmonds' problem is equivalent to checking if the following determinantal polynomial P A (x 1 , ..., x k ) = det(is not identically equal to zero. This determinantal polynomial can be efficiently evaluated, hence randomized poly-time algorithms, based on Schwartz's lemma or its recent improvements, are readily available (notice that our problem is defined over infinite field with infinite characteristic). But for general linear subspaces of M(N), i.e. without an extra assumption (promise), poly-time deterministic algorithms are not known and the problem is believed to be "HARD" . Like any other homogeneous polynomial, P A (x 1 , ..., x k ) is a weighted sum of monomials of degree N , i.e.where I k,N stands for a set of vectors r = (r 1 , ..., r k ) with nonnegative integer components and 1≤i≤k r i = N . We will make substantial use of the following (Hilbert) norm of determinantal polynomial P (.) : P 2 G =:It is easy to show tha...

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Section: Suppose That Linear Subspacesmentioning

confidence: 99%

Classical deterministic complexity of Edmonds' Problem and quantum entanglement

Gurvits

2003

Proceedings of the Thirty-Fifth Annual ACM Symposium on Theory of Computing

390

354

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“…The Coset Intersection problem for permutation groups of degree n can be solved in quasi-polynomial (exp((log n) O(1) )) time [Bab16], while a relatively easier singly exponential (exp(O(n))) algorithm has been obtained by Luks [Luk99]. Algorithms over finite-dimensional algebras have been considered in, e. g., [CIK97,BL08,IKS10]. In particular, over finite fields, polynomial-time algorithms for Module Isomorphism and Module Cyclicity are first shown in [CIK97].…”

Section: Preliminaries For the Algorithmsmentioning

confidence: 99%

“…Algorithms over finite-dimensional algebras have been considered in, e. g., [CIK97,BL08,IKS10]. In particular, over finite fields, polynomial-time algorithms for Module Isomorphism and Module Cyclicity are first shown in [CIK97].…”

Section: Preliminaries For the Algorithmsmentioning

confidence: 99%

Algorithms for Group Isomorphism via Group Extensions and Cohomology

Grochow

Qiao

2014

2014 IEEE 29th Conference on Computational Complexity (CCC)

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The isomorphism problem for finite groups of order n (GpI) has long been known to be solvable in n log n+O(1) time, but only recently were polynomial-time algorithms designed for several interesting group classes. Inspired by recent progress, we revisit the strategy for GpI via the extension theory of groups.The extension theory describes how a normal subgroup N is related to G/N via G, and this naturally leads to a divide-and-conquer strategy that "splits" GpI into two subproblems: one regarding group actions on other groups, and one regarding group cohomology. When the normal subgroup N is abelian, this strategy is well-known. Our first contribution is to extend this strategy to handle the case when N is not necessarily abelian. This allows us to provide a unified explanation of all recent polynomial-time algorithms for special group classes.Guided by this strategy, to make further progress on GpI, we consider central-radical groups, proposed in Babai et al. (SODA 2011): the class of groups such that G mod its center has no abelian normal subgroups. This class is a natural extension of the group class considered by Babai et al. (ICALP 2012), namely those groups with no abelian normal subgroups. Following the above strategy, we solve GpI in n O(log log n) time for central-radical groups, and in polynomial time for several prominent subclasses of central-radical groups. We also solve GpI in n O(log log n) time for groups whose solvable normal subgroups are elementary abelian but not necessarily central. As far as we are aware, this is the first time there have been worstcase guarantees on a n o(log n) -time algorithm that tackles both aspects of GpI-actions and cohomology-simultaneously.Prior to this work, the best proven upper bounds on algorithms for groups with central radicals were n O(log n) , even for groups with a central radical of constant size, such as Rad(G) = Z(G) = Z 2 . To develop our new algorithms we utilize several mathematical results on the detailed structure of cohomology classes, as well as algorithmic results for code equivalence, coset intersection and cyclicity testing of modules over finite-dimensional associative algebras. We also suggest several promising directions for future work. * The introduction may serve as an extended abstract:

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“…The MSCP in matrix group is also easy because the equation AX = XC can be considered as a system of homogeneous linear equations in the entries of X. One can use the polynomial time deterministic algorithm by Chistov, Ivanyos, and Karpinski [7].…”

Section: Theoremmentioning

confidence: 99%

“…. , l m ) are m-cycles in a i such that z(k 1 ) = l 1 , then z(k j ) = l j for all 2 ≤ j ≤ m. For example, let a 1 = (1, 2)(4, 5)(8, 9), a 2 = (3, 4) (5,7,8), and a 3 = (1, 5, 6)(9, 10) be the cycle decompositions of permutations in S 10 . Let z ∈ ∩ 3 i=1 Cent(a i ).…”

Section: Theoremmentioning

confidence: 99%

Potential Weaknesses of the Commutator Key Agreement Protocol Based on Braid Groups

Lee¹,

Lee

2002

Advances in Cryptology — EUROCRYPT 2002

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Abstract. The braid group with its conjugacy problem is one of the recent hot issues in cryptography. At CT-RSA 2001, Anshel, Anshel, Fisher, and Goldfeld proposed a commutator key agreement protocol (KAP) based on the braid groups and their colored Burau representation. Its security is based on the multiple simultaneous conjugacy problem (MSCP) plus a newly adopted key extractor. This article shows how to reduce finding the shared key of this KAP to the list-MSCPs in a permutation group and in a matrix group over a finite field. We also develop a mathematical algorithm for the MSCP in braid groups. The former implies that the usage of colored Burau representation in the key extractor causes a new weakness, and the latter can be used as a tool to investigate the security level of their KAP.

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Polynomial time algorithms for modules over finite dimensional algebras

Cited by 50 publications

References 14 publications

Classical deterministic complexity of Edmonds' Problem and quantum entanglement

Classical deterministic complexity of Edmonds' Problem and quantum entanglement

Algorithms for Group Isomorphism via Group Extensions and Cohomology

Potential Weaknesses of the Commutator Key Agreement Protocol Based on Braid Groups

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