The status of polycyclic group-based cryptography: A survey and open problems
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Jonathan Gryak
Abstract
Polycyclic groups are natural generalizations of cyclic groups but with more complicated algorithmic properties. They are finitely presented and the word, conjugacy, and isomorphism decision problems are all solvable in these groups. Moreover, the non-virtually nilpotent ones exhibit an exponential growth rate. These properties make them suitable for use in group-based cryptography, which was proposed in 2004 by Eick and Kahrobaei [10]. Since then, many cryptosystems have been created that employ polycyclic groups. These include key exchanges such as non-commutative ElGamal, authentication schemes based on the twisted conjugacy problem, and secret sharing via the word problem. In response, heuristic and deterministic methods of cryptanalysis have been developed, including the length-based and linear decomposition attacks. Despite these efforts, there are classes of infinite polycyclic groups that remain suitable for cryptography. The analysis of algorithms for search and decision problems in polycyclic groups has also been developed. In addition to results for the aforementioned problems we present those concerning polycyclic representations, group morphisms, and orbit decidability. Though much progress has been made, many algorithmic and complexity problems remain unsolved; we conclude with a number of them. Of particular interest is to show that cryptosystems using infinite polycyclic groups are resistant to cryptanalysis on a quantum computer.
Funding source: National Science Foundation
Award Identifier / Grant number: CCF-1564968
Funding source: Office of Naval Research
Award Identifier / Grant number: N00014-15-1-2164
Funding statement: Delaram Kahrobaei is partially supported by a PSC-CUNY grant from the CUNY Research Foundation, the City Tech Foundation, and ONR (Office of Naval Research) grant N00014-15-1-2164. Delaram Kahrobaei has also partially supported by an NSF travel grant CCF-1564968 to IHP in Paris.
Acknowledgements
We would like to thank Bettina Eick for her contributions regarding polycyclic groups and their algorithmic properties.
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© 2016 by De Gruyter
Articles in the same Issue
- Frontmatter
- Computing discrete logarithms using 𝒪((log q)2) operations from {+,-,×,÷,&}
- A parallel evolutionary approach to solving systems of equations in polycyclic groups
- Authenticated commutator key agreement protocol
- On the covering number of small symmetric groups and some sporadic simple groups
- Compositions of linear functions and applications to hashing
- Hydra group doubles are not residually finite
- The status of polycyclic group-based cryptography: A survey and open problems
- On irreducible algebraic sets over linearly ordered semilattices
- A nonlinear decomposition attack
Articles in the same Issue
- Frontmatter
- Computing discrete logarithms using 𝒪((log q)2) operations from {+,-,×,÷,&}
- A parallel evolutionary approach to solving systems of equations in polycyclic groups
- Authenticated commutator key agreement protocol
- On the covering number of small symmetric groups and some sporadic simple groups
- Compositions of linear functions and applications to hashing
- Hydra group doubles are not residually finite
- The status of polycyclic group-based cryptography: A survey and open problems
- On irreducible algebraic sets over linearly ordered semilattices
- A nonlinear decomposition attack
