Search and test algorithms for triple product property triples
-
Ivo Hedtke
and
Sandeep Murthy
Abstract.
In 2003 Cohn and Umans introduced a group-theoretic approach to
fast matrix multiplication.
This involves finding large subsets of a group 
satisfying the triple product
property (TPP) as a means to bound the exponent 
of matrix multiplication.
We present two new characterizations of the TPP, which are used for theoretical
considerations and for TPP test algorithms.
We describe the algorithms for all known TPP tests and present the runtime
differences between their GAP implementations.
We prove that the search for non-trivial-sized TPP triples of subgroups of a given
group can be restricted to the set of its non-normal subgroups, and apply this,
together with other preconditions, to describe brute-force search algorithms for
largest-sized TPP triples of subgroups and subsets.
In addition we present the results of the subset brute-force search for all groups of
order up to 32 and selected results of the subgroup brute-force search for
2-groups, 
and 
.
Our results for the groups 
and 
suggest tentative
answers to certain questions posed by Cohn and Umans.
© 2012 by Walter de Gruyter Berlin Boston
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- Masthead
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- Continuous hard-to-invert functions and biometric authentication
- Existence, algorithms, and asymptotics of direct product decompositions, I
- Isomorphism in expanding families of indistinguishable groups
- Search and test algorithms for triple product property triples
- Evolutionary algorithm solution of the multiple conjugacy search problem in groups, and its applications to cryptography
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Articles in the same Issue
- Masthead
- Two-party key establishment: From passive to active security without introducing new assumptions
- Continuous hard-to-invert functions and biometric authentication
- Existence, algorithms, and asymptotics of direct product decompositions, I
- Isomorphism in expanding families of indistinguishable groups
- Search and test algorithms for triple product property triples
- Evolutionary algorithm solution of the multiple conjugacy search problem in groups, and its applications to cryptography
- A Diffie–Hellman key exchange protocol using matrices over noncommutative rings
- No-leak authentication by the Sherlock Holmes method
