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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 10 (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
PolynomialTime Normalizers for Permutation Groups With Restricted Composition Factors
, 2002
"... For an integer constant d > 0, let d denote the class of finite groups all of whose nonabelian composition factors lie in S d ; in particular, d includes all solvable groups. Motivated by applications to graphisomorphism testing, there has been extensive study of the complexity of computation fo ..."
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Cited by 4 (1 self)
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For an integer constant d > 0, let d denote the class of finite groups all of whose nonabelian composition factors lie in S d ; in particular, d includes all solvable groups. Motivated by applications to graphisomorphism testing, there has been extensive study of the complexity of computation for permutation groups in this class. In particular, setstabilizers, group intersections, and centralizers have all been shown to be polynomialtime computable. The most notable gap in the theory has been the question of whether normalizers of subgroups can be found in polynomial time. We resolve this question in the affirmative. Among other new procedures, the algorithm requires instances of subspacestabilizers for certain linear representations and therefore some polynomialtime computation in matrix groups.
Deterministic algorithms for management of matrix groups, in: Groups and computation
, 1999
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Polynomialtime normalizers
"... For an integer constant d>0, let Γd denote the class of finite groups all of whose nonabelian composition factors lie in Sd; in particular, Γd includes all solvable groups. Motivated by applications to graphisomorphism testing, there has been extensive study of the complexity of computation for ..."
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For an integer constant d>0, let Γd denote the class of finite groups all of whose nonabelian composition factors lie in Sd; in particular, Γd includes all solvable groups. Motivated by applications to graphisomorphism testing, there has been extensive study of the complexity of computation for permutation groups in this class. In particular, the problems of finding set stabilizers, intersections and centralizers have all been shown to be polynomialtime computable. A notable open issue for the class Γd has been the question of whether normalizers can be found in polynomial time. We resolve this question in the affirmative. We prove that, given permutation groups G, H ≤ Sym(Ω) such that G ∈ Γd, the normalizer of H in G can be found in polynomial time. Among other new procedures, our method includes a key subroutine to solve the problem of finding stabilizers of subspaces in linear representations of permutation groups in Γd.