Algebraic cryptography: new constructions and their security against provable break
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D. Grigoriev, A. Kojevnikov and S. J. Nikolenko
Translated by: the authors - St. Petersburg Math. J. 20 (2009), 937-953
- DOI: https://doi.org/10.1090/S1061-0022-09-01079-6
- Published electronically: October 1, 2009
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Abstract:
Very few known cryptographic primitives are based on noncommutative algebra. Each new scheme is of substantial interest, because noncommutative constructions are secure against many standard cryptographic attacks. On the other hand, cryptography does not provide security proofs that might allow the security of a cryptographic primitive to rely upon structural complexity assumptions. Thus, it is important to investigate weaker notions of security.
In this paper, new constructions of cryptographic primitives based on group invariants are proposed, together with new ways to strengthen them for practical use. Also, the notion of a provable break is introduced, which is a weaker version of the regular cryptographic break. In this new version, an adversary should have a proof that he has correctly decyphered the message. It is proved that the cryptosystems based on matrix group invariants and a version of the Anshel–Anshel–Goldfeld key agreement protocol for modular groups are secure against provable break unless $\mathrm {NP}=\mathrm {RP}$.
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Bibliographic Information
- D. Grigoriev
- Affiliation: CNRS, Laboratoire des Mathématiques, Université de Lille, 59655 Villeneuve d’Ascq, France
- Email: Dmitry.Grigoryev@math.univ-lille1.fr
- A. Kojevnikov
- Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, 191023 St. Petersburg, Russia
- Email: arist@pdmi.ras.ru
- S. J. Nikolenko
- Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, 191023 St. Petersburg, Russia
- Email: sergey@logic.pdmi.ras.ru
- Received by editor(s): January 9, 2008
- Published electronically: October 1, 2009
- Additional Notes: The research was done during the stay at the Max-Planck-Institut für Mathematik, Bonn, Germany.
The second and third authors were supported in part by INTAS (YSF fellowship no. 05-109-5565) and by RFBR (grant nos. 05-01-00932 and 06-01-00502). - © Copyright 2009 American Mathematical Society
- Journal: St. Petersburg Math. J. 20 (2009), 937-953
- MSC (2000): Primary 94A60, 68P25, 11T71
- DOI: https://doi.org/10.1090/S1061-0022-09-01079-6
- MathSciNet review: 2530896