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Polynomial time solutions of computational problems in noncommutativealgebraic cryptography
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Polynomial time cryptanalysis of noncommutativealgebraic key exchange protocols
"... We introduce the linear centralizer method for a passive adversary to extract the shared key in grouptheory based key exchange protocols (KEPs). We apply this method to obtain a polynomial time cryptanalysis of the Commutator KEP, introduced by Anshel–Anshel–Goldfeld in 1999 and considered extensiv ..."
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We introduce the linear centralizer method for a passive adversary to extract the shared key in grouptheory based key exchange protocols (KEPs). We apply this method to obtain a polynomial time cryptanalysis of the Commutator KEP, introduced by Anshel–Anshel–Goldfeld in 1999 and considered extensively ever since. We also apply this method to the Centralizer KEP, introduced by Shpilrain–Ushakov in 2006. Our method is proved to be of polynomial time using a technical lemma about sampling invertible matrices from a linear space of matrices.
Double cosets for SU(2) × · · · × SU(2) and outer automorphisms of free groups
, 903
"... Consider the space of double cosets of the product of n copies of SU(2) with respect to the diagonal subgroup. We get a parametrization of this space, the radial part of the Haar measure, and explicit formulas for the actions of the group of outer automorphisms of the free group Fn−1 and of the brai ..."
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Consider the space of double cosets of the product of n copies of SU(2) with respect to the diagonal subgroup. We get a parametrization of this space, the radial part of the Haar measure, and explicit formulas for the actions of the group of outer automorphisms of the free group Fn−1 and of the braid group of n − 1 strings. 1
GROUP THEORY IN CRYPTOGRAPHY
, 2009
"... This paper is a guide for the pure mathematician who would like to know more about cryptography based on group theory. The paper gives a brief overview of the subject, and provides pointers to good textbooks, key research papers and recent survey papers in the area. ..."
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This paper is a guide for the pure mathematician who would like to know more about cryptography based on group theory. The paper gives a brief overview of the subject, and provides pointers to good textbooks, key research papers and recent survey papers in the area.