Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Groups Complexity Cryptology

Managing Editor: Shpilrain, Vladimir / Weil, Pascal

Editorial Board Member: Conder, Marston / Dehornoy, Patrick / Eick, Bettina / Fine, Benjamin / Gilman, Robert / Grigoriev, Dima / Ko, Ki Hyoung / Kreuzer, Martin / Mikhalev, Alexander V. / Myasnikov, Alexei / Perret, Ludovic / Roman'kov, Vitalii / Rosenberger, Gerhard / Sapir, Mark / Thomas, Rick / Tsaban, Boaz / Capell, Enric Ventura

2 Issues per year


CiteScore 2016: 0.35

SCImago Journal Rank (SJR) 2016: 0.372
Source Normalized Impact per Paper (SNIP) 2016: 0.517

Mathematical Citation Quotient (MCQ) 2016: 0.23

Online
ISSN
1869-6104
See all formats and pricing
More options …

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

Jonathan Gryak / 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
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2016-10-11 | DOI: https://doi.org/10.1515/gcc-2016-0013

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.

Keywords: Polycyclic groups; cryptography; complexity

MSC 2010: 94A60; 20F10

References

  • [1]

    Anshel I., Anshel M. and Goldfeld D., An algebraic method for public-key cryptography, Math. Res. Lett. 6 (1999), 287–291. Google Scholar

  • [2]

    Assmann B. and Linton S., Using the Mal’cev correspondence for collection in polycyclic groups, J. Algebra 316 (2007), no. 2, 828–848. Google Scholar

  • [3]

    Auslander L., The automorphism group of a polycyclic group, Ann. of Math. (2) 89 (1969), 314–322. Google Scholar

  • [4]

    Batty M., Rees S., Braunstein S. and Duncan A., Quantum algorithms in group theory, Computational and Experimental Group Theory (Baltimore 2003), Contemp. Math. 349, American Mathematical Society, Providence (2004), 1–62. Google Scholar

  • [5]

    Bogopolski O., Martino A. and Ventura E., Orbit decidability and the conjugacy problem for some extensions of groups, Trans. Amer. Math. Soc. 362 (2010), no. 4, 2003–2036. Google Scholar

  • [6]

    Bonanome M., Quantum algorithms in combinatorial group theory, Ph.D. thesis, City University of New York, 2007. Google Scholar

  • [7]

    Dehn M., Über unendliche diskontinuierliche Gruppen, Math. Ann. 71 (1911), no. 1, 116–144. Google Scholar

  • [8]

    du Sautoy M., Polycyclic groups, analytic groups and algebraic groups, Proc. Lond. Math. Soc. (3) 85 (2002), no. 1, 62–92. Google Scholar

  • [9]

    Eick B., When is the automorphism group of a virtually polycyclic group virtually polycyclic?, Glasg. Math. J. 45 (2003), no. 3, 527–533. Google Scholar

  • [10]

    Eick B. and Kahrobaei D., Polycyclic groups: A new platform for cryptography, preprint 2004, http://arxiv.org/abs/math/0411077.

  • [11]

    Eick B. and Ostheimer G., On the orbit-stabilizer problem for integral matrix actions of polycyclic groups, Math. Comp. 72 (2003), no. 243, 1511–1529. Google Scholar

  • [12]

    Fesenko A., Vulnerability of cryptographic primitives based on the power conjugacy search problem in quantum computing, Cybernet. Systems Anal. 50 (2014), no. 5, 815–816. Google Scholar

  • [13]

    Formanek E., Conjugate separability in polycyclic groups, J. Algebra 42 (1976), no. 1, 1–10. Google Scholar

  • [14]

    Garber D., Kahrobaei D. and Lam H. T., Length-based attack for polycyclic groups, J. Math. Cryptol. 9 (2015), 33–44. Google Scholar

  • [15]

    Garber D., Kaplan S., Teicher M., Tsaban B. and Vishne U., Length-based conjugacy search in the braid group, Algebraic Methods in Cryptography (Bochum/Mainz 2005), Contemp. Math. 418, American Mathematical Society, Providence (2006), 75–87. Google Scholar

  • [16]

    Gebhardt V., Efficient collection in infinite polycyclic groups, J. Symbolic Comput. 34 (2002), no. 3, 213–228. Google Scholar

  • [17]

    Grigoriev D. and Shpilrain V., Zero-knowledge authentication schemes from actions on graphs, groups, or rings, Ann. Pure Appl. Logic 162 (2010), 194–200. Google Scholar

  • [18]

    Habeeb M., Kahrobaei D. and Shpilrain V., A secret sharing scheme based on group presentations and the word problem, Computational and Combinatorial Group Theory and Cryptography (Las Vegas/Ithaca 2011), Contemp. Math. 582, American Mathematical Society, Providence (2012), 143–150. Google Scholar

  • [19]

    Hall P., The Edmonton Notes on Nilpotent Groups, Queen Mary College Math. Notes, Queen Mary College, London, 1969. Google Scholar

  • [20]

    Holt D. F., Eick B. and O’Brien E. A., Handbook of Computational Group Theory, Discrete Math. Appl. (Boca Raton), Chapman & Hall/CRC, Boca Raton, 2005. Google Scholar

  • [21]

    Hughes J. and Tannenbaum A., Length-based attacks for certain group based encryption rewriting systems, preprint 2003, https://arxiv.org/abs/cs/0306032.

  • [22]

    Ivanyos G., Sanselme L. and Santha M., An efficient quantum algorithm for the hidden subgroup problem in nil-2 groups, LATIN 2008 – Theoretical Informatics (Buzios 2008), Lecture Notes in Comput. Sci. 4957, Springer, Berlin (2008), 759–771. Google Scholar

  • [23]

    Kahrobaei D. and Khan B., Nis05-6: A non-commutative generalization of ElGamal key exchange using polycyclic groups, IEEE Global Telecommunications Conference (GLOBECOM ’06), IEEE Press, Piscataway (2006), 1–5. Google Scholar

  • [24]

    Kahrobaei D. and Koupparis C., Non-commutative digital signatures using non-commutative groups, Groups Complex. Cryptol. 4 (2012), 377–384. Google Scholar

  • [25]

    Ko K. H., Lee S. J., Cheon J. H., Han J. W., Kang J. and Park C., New public-key cryptosystem using braid groups, Advances in Cryptology (CRYPTO 2000), Lecture Notes in Comput. Sci. 1880, Springer, Berlin (2000), 166–183. Google Scholar

  • [26]

    Kotov M. and Ushakov A., Analysis of a certain polycyclic-group-based cryptosystem, J. Math. Cryptol. 9 (2015), no. 3, 161–167. Google Scholar

  • [27]

    Leedham-Green C. R. and Soicher L. H., Collection from the left and other strategies, J. Symbolic Comput. 9 (1990), no. 5–6, 665–675. Google Scholar

  • [28]

    Lo E. and Ostheimer G., A practical algorithm for finding matrix representations for polycyclic groups, J. Symbolic Comput. 28 (1999), no. 3, 339–360. Google Scholar

  • [29]

    Mal’cev A., On homomorphisms onto finite groups, Trans. Amer. Math. Soc. 119 (1983), 67–79. Google Scholar

  • [30]

    Milnor J., Growth of finitely generated solvable groups, J. Differential Geom. 2 (1968), no. 4, 447–449. Google Scholar

  • [31]

    Myasnikov A. D. and Ushakov A., Length-based attack and braid groups: Cryptanalysis of Anshel–Anshel–Goldfeld key-exchange protocol, Public Key Cryptography – PKC 2007 (Beijing 2007), Lecture Notes in Comput. Sci. 4450, Springer, Berlin (2007), 76–88. Google Scholar

  • [32]

    Myasnikov A. G. and Roman’kov V., A linear decomposition attack, Groups Complex. Cryptol. 7 (2015), no. 1, 81–94. Google Scholar

  • [33]

    Myasnikov A. G., Shpilrain V., Ushakov A. and Mosina N., Non-Commutative Cryptography and Complexity of Group-Theoretic Problems, Math. Surveys Monogr. 177, American Mathematical Society, Providence, 2011. Google Scholar

  • [34]

    Myasnikov A. G. and Ushakov A., Random subgroups and analysis of the length-based and quotient attacks, J. Math. Cryptol. 2 (2008), no. 1, 29–61. Google Scholar

  • [35]

    Nickel W., Matrix representations for torsion-free nilpotent groups by Deep Thought, J. Algebra 300 (2006), no. 1, 376–383. Google Scholar

  • [36]

    Remeslennikov V., Conjugacy in polycyclic groups, Algebra Logic 8 (1969), no. 6, 404–411. Google Scholar

  • [37]

    Roman’kov V., The twisted conjugacy problem for endomorphisms of polycyclic groups, J. Group Theory 13 (2010), no. 3, 355–364. Google Scholar

  • [38]

    Segal D., Decidable properties of polycyclic groups, Proc. Lond. Math. Soc. (3) 61 (1990), no. 3, 61–497. Google Scholar

  • [39]

    Shor P., Algorithms for quantum computation: Discrete logarithms and factoring, 35th Annual Symposium on Foundations of Computer Science, IEEE Press, Piscataway (1994), 124–134. Google Scholar

  • [40]

    Shpilrain V., Search and witness problems in group theory, Groups Complex. Cryptol. 2 (2010), no. 2, 231–246. Google Scholar

  • [41]

    Shpilrain V. and Ushakov A., The conjugacy search problem in public key cryptography: Unnecessary and insufficient, Appl. Algebra Engrg. Comm. Comput. 17 (2006), no. 3–4, 285–289. Google Scholar

  • [42]

    Shpilrain V. and Ushakov A., An authentication scheme based on the twisted conjugacy problem, Applied Cryptography and Network Security, Lecture Notes in Comput. Sci. 5037, Springer, Berlin (2008), 366–372. Google Scholar

  • [43]

    Shpilrain V. and Zapata G., Using the subgroup membership search problem in public key cryptography, Algebraic Methods in Cryptography (Bochum/Mainz 2005), Contemp. Math. 418, American Mathematical Society, Providence (2006), 169–178. Google Scholar

  • [44]

    Tsaban B., Polynomial-time solutions of computational problems in noncommutative-algebraic cryptography, J. Cryptology 28 (2015), 601–622. Google Scholar

  • [45]

    Wehrfritz B., Two remarks on polycyclic groups, Bull. Lond. Math. Soc. 26 (1994), no. 6, 543–548. Google Scholar

  • [46]

    Wolf J., Growth of finitely generated solvable groups and curvature of Riemannian manifolds, J. Differential Geom. 2 (1968), 421–446. Google Scholar

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

© 2016 by De Gruyter. Copyright Clearance Center

Comments (0)

Please log in or register to comment.
Log in