Efficient Algorithms for Solving Overdefined Systems of Multivariate Polynomial Equations
 Nicolas Courtois,
 Alexander Klimov,
 Jacques Patarin,
 Adi Shamir
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Abstract
The security of many recently proposed cryptosystems is based on the difficulty of solving large systems of quadratic multivariate polynomial equations. This problem is NPhard over any field. When the number of equations m is the same as the number of unknowns n the best known algorithms are exhaustive search for small fields, and a Gröbner base algorithm for large fields. Gröbner base algorithms have large exponential complexity and cannot solve in practice systems with n ≥ 15. Kipnis and Shamir [9] have recently introduced a new algorithm called “relinearization”. The exact complexity of this algorithm is not known, but for sufficiently overdefined systems it was expected to run in polynomial time.
In this paper we analyze the theoretical and practical aspects of relinearization. We ran a large number of experiments for various values of n and m, and analysed which systems of equations were actually solvable. We show that many of the equations generated by relinearization are linearly dependent, and thus relinearization is less efficient that one could expect. We then develop an improved algorithm called XL which is both simpler and more powerful than relinearization. For all 0 < ε ≤ 1/2, and m ≥ εn ^{2}, XL and relinearization are expected to run in polynomial time of approximately \( n^{\mathcal{O}(1/\sqrt \varepsilon )} \) . Moreover, we provide strong evidence that relinearization and XL can solve randomly generated systems of polynomial equations in subexponential time when m exceeds n by a number that increases slowly with n.
 Iyad A. Ajwa, Zhuojun Liu, and Paul S. Wang: “Grobner Bases Algorithm”, ICM Technical Reports, February 1995. See http://symbolicnet.mcs.kent.edu/icm/reports/index1995.html.
 Don Coppersmith: “Finding a small root of a univariate modular equation”; Proceedings of Eurocrypt’96, SpringerVerlag, pp.155–165.
 Nicolas Courtois “The security of HFE”, to be published.
 Nicolas Courtois: The HFE cryptosystem web page. See http://www.univtln.fr/~courtois/hfe.html
 JeanCharles Faugère: “A new efficient algorithm for computing Gröbner bases (F_{4}).” Journal of Pure and Applied Algebra 139 (1999) pp. 61–88. See http://www.elsevier.com/locate/jpaa CrossRef
 JeanCharles Faugère: “Computing Gröbner basis without reduction to 0”, technical report LIP6, in preparation, source: private communication.
 Rudolf Lidl, Harald Niederreiter: “Finite Fields”; Encyclopedia of Mathematics and its applications, Volume 20, Cambridge University Press.
 Aviad Kipnis, Jacques Patarin, Louis Goubin: “Unbalanced Oil and Vinegar Signature Schemes”; Eurocrypt 1999, SpringerVerlag, pp. 216–222.
 Aviad Kipnis, Adi Shamir: “Cryptanalysis of the HFE Public Key Cryptosystem”; Proceedings of Crypto’99, SpringerVerlag.
 Neal Koblitz: “Algebraic aspects of cryptography”; SpringerVerlag, ACM3, 1998, Chapter 4 “Hidden Monomial Cryptosystems”, pp. 80–102.
 Jacques Patarin: “Hidden Field Equations (HFE) and Isomorphisms of Polynomials (IP): two new families of Asymmetric Algorithms”; Eurocrypt’96, Springer Verlag, pp. 33–48. An extended uptodate version can be found at http://www.univtln.fr/~courtois/hfe.ps
 Title
 Efficient Algorithms for Solving Overdefined Systems of Multivariate Polynomial Equations
 Book Title
 Advances in Cryptology — EUROCRYPT 2000
 Book Subtitle
 International Conference on the Theory and Application of Cryptographic Techniques Bruges, Belgium, May 14–18, 2000 Proceedings
 Pages
 pp 392407
 Copyright
 2000
 DOI
 10.1007/3540455396_27
 Print ISBN
 9783540675174
 Online ISBN
 9783540455394
 Series Title
 Lecture Notes in Computer Science
 Series Volume
 1807
 Series ISSN
 03029743
 Publisher
 Springer Berlin Heidelberg
 Copyright Holder
 SpringerVerlag Berlin Heidelberg
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 Editors

 Bart Preneel ^{(4)}
 Editor Affiliations

 4. Department of Electrical Engineering  ESAT / COSIC, Katholieke Universiteit Leuven
 Authors

 Nicolas Courtois ^{(5)} ^{(7)}
 Alexander Klimov ^{(6)}
 Jacques Patarin ^{(7)}
 Adi Shamir ^{(8)}
 Author Affiliations

 5. MS/LI, Toulon University, BP 132, F83957, La Garde Cedex, France
 7. Bull CP8, 68, route de Versailles, BP45, 78431, Louveciennes Cedex, France
 6. Dept. of Appl. Math. & Cybernetics, Moscow State University, Moscow, Russia
 8. Dept. of Applied Math., The Weizmann Institute of Science, Rehovot, 76100, Israel
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