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Groups Complexity Cryptology

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The status of polycyclic group-based cryptography: A survey and open problems

Jonathan Gryak
  • CUNY Graduate Center, PhD Program in Computer Science, City University of New York, United States of America
  • Email:
/ Delaram Kahrobaei
  • Corresponding author
  • CUNY Graduate Center, PhD Program in Computer Science and NYCCT, Mathematics Department, City University of New York, United States of America
  • Email:
Published Online: 2016-10-11 | DOI: https://doi.org/10.1515/gcc-2016-0013


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.

Keywords: Polycyclic groups; cryptography; complexity

MSC 2010: 94A60; 20F10


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About the article

Received: 2016-06-22

Published Online: 2016-10-11

Published in Print: 2016-11-01

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

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.

Citation Information: Groups Complexity Cryptology, ISSN (Online) 1869-6104, ISSN (Print) 1867-1144, DOI: https://doi.org/10.1515/gcc-2016-0013. Export Citation

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