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Journal of Mathematical Cryptology

Managing Editor: Magliveras, Spyros S. / Steinwandt, Rainer / Trung, Tran

Editorial Board: Blackburn, Simon R. / Blundo, Carlo / Burmester, Mike / Cramer, Ronald / Dawson, Ed / Gilman, Robert / Gonzalez Vasco, Maria Isabel / Grosek, Otokar / Helleseth, Tor / Kim, Kwangjo / Koblitz, Neal / Kurosawa, Kaoru / Lauter, Kristin / Lange, Tanja / Menezes, Alfred / Nguyen, Phong Q. / Pieprzyk, Josef / Rötteler, Martin / Safavi-Naini, Rei / Shparlinski, Igor E. / Stinson, Doug / Takagi, Tsuyoshi / Williams, Hugh C. / Yung, Moti

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1862-2984
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Volume 3, Issue 3

Hybrid approach for solving multivariate systems over finite fields

Luk Bettale
  • INRIA, Centre Paris-Rocquencourt, SALSA Project, UPMC, Univ Paris 06, LIP6, CNRS, UMR 7606, LIP6, 104, avenue du Président Kennedy, 75016 Paris, France. Email:
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Jean-Charles Faugère
  • INRIA, Centre Paris-Rocquencourt, SALSA Project, UPMC, Univ Paris 06, LIP6, CNRS, UMR 7606, LIP6, 104, avenue du Président Kennedy, 75016 Paris, France. Email:
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Ludovic Perret
  • INRIA, Centre Paris-Rocquencourt, SALSA Project, UPMC, Univ Paris 06, LIP6, CNRS, UMR 7606, LIP6, 104, avenue du Président Kennedy, 75016 Paris, France. Email:
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2010-01-20 | DOI: https://doi.org/10.1515/JMC.2009.009

Abstract

In this paper, we present an improved approach to solve multivariate systems over finite fields. Our approach is a tradeoff between exhaustive search and Gröbner bases techniques. We give theoretical evidences that our method brings a significant improvement in a very large context and we clearly define its limitations. The efficiency depends on the choice of the tradeoff. Our analysis gives an explicit way to choose the best tradeoff as well as an approximation. From our analysis, we present a new general algorithm to solve multivariate polynomial systems. Our theoretical results are experimentally supported by successful cryptanalysis of several multivariate schemes (TRMS, UOV, . . . ). As a proof of concept, we were able to break the proposed parameters assumed to be secure until now. Parameters that resists to our method are also explicitly given. Our work permits to refine the parameters to be chosen for multivariate schemes.

Keywords.: Gröbner bases; multivariate cryptography

About the article

Received: 2008-12-30

Revised: 2009-10-12

Published Online: 2010-01-20

Published in Print: 2009-09-01


Citation Information: Journal of Mathematical Cryptology, Volume 3, Issue 3, Pages 177–197, ISSN (Online) 1862-2984, ISSN (Print) 1862-2976, DOI: https://doi.org/10.1515/JMC.2009.009.

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