Advances in Cryptology — EUROCRYPT'98
Volume 1403 of the series Lecture Notes in Computer Science pp 184200
Improved algorithms for isomorphisms of polynomials
 Jacques PatarinAffiliated withBull Smart Cards and Terminals
 , Louis GoubinAffiliated withBull Smart Cards and Terminals
 , Nicolas CourtoisAffiliated withModélisation et Signal, Université de Toulon et du Var
Abstract
This paper is about the design of improved algorithms to solve Isomorphisms of Polynomials (IP) problems. These problems were first explicitly related to the problem of finding the secret key of some asymmetric cryptographic algorithms (such as Matsumoto and Imai's C* scheme of [12], or some variations of Patarin's HFE scheme of [14]). Moreover, in [14], it was shown that IP can be used in order to design an asymmetric authentication or signature scheme in a straightforward way. We also introduce the more general Morphisms of Polynomials problem (MP). As we see in this paper, these problems IP and MP have deep links with famous problems such as the Isomorphism of Graphs problem or the problem of fast multiplication of n x n matrices. The complexities of our algorithms for IP are still not polynomial, but they are much more efficient than the previously known algorithms. For example, for the IP problem of finding the two secret matrices of a MatsumotoImai C* scheme over K = Fq, the complexity of our algorithms is \(\mathcal{O}(q^{n/2} )\) instead of \(\mathcal{O}(q^{(n^2 )} )\) for previous algorithms. (In [13], the C* scheme was broken, but the secret key was not found). Moreover, we have algorithms to achieve a complexity \(\mathcal{O}(q^{\tfrac{3}{2}n} )\) on any system of n quadratic equations with n variables over K = Fq (not only equations from C*). We also show that the problem of deciding whether a polynomial isomorphism exists between two sets of equations is not NPcomplete (assuming the classical hypothesis about ArthurMerlin games), but solving IP is at least as difficult as the Graph Isomorphism problem (GI) (and perhaps much more difficult), so that IP is unlikely to be solvable in polynomial time. Moreover, the more general Morphisms of Polynomials problem (MP) is NPhard. Finally, we suggest some variations of the IP problem that may be particularly convenient for cryptographic use.
 Title
 Improved algorithms for isomorphisms of polynomials
 Book Title
 Advances in Cryptology — EUROCRYPT'98
 Book Subtitle
 International Conference on the Theory and Application of Cryptographic Techniques Espoo, Finland, May 31 – June 4, 1998 Proceedings
 Pages
 pp 184200
 Copyright
 1998
 DOI
 10.1007/BFb0054126
 Print ISBN
 9783540645184
 Online ISBN
 9783540697954
 Series Title
 Lecture Notes in Computer Science
 Series Volume
 1403
 Series ISSN
 03029743
 Publisher
 Springer Berlin Heidelberg
 Copyright Holder
 SpringerVerlag
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 Editors
 Authors

 Jacques Patarin ^{(1)}
 Louis Goubin ^{(1)}
 Nicolas Courtois ^{(2)}
 Author Affiliations

 1. Bull Smart Cards and Terminals, 68 route de Versailles, BP 45, 78431, Louveciennes Cedex, France
 2. Modélisation et Signal, Université de Toulon et du Var, BP 132, 83957, La Garde Cedex, France
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