Abstract
The threat quantum computing poses to traditional cryptosystems (such as RSA, elliptic-curve cryptosystems) has brought about the appearance of new systems resistant to it: among them, multivariate quadratic public-key ones. The security of the latter kind of cryptosystems is related to the isomorphism of polynomials (IP) problem. In this work, we study some aspects of the equivalence relation the IP problem induces over the set of quadratic polynomial maps and the determination of its equivalence classes. We contribute two results. First, we prove that when determining these classes, it suffices to consider the affine transformation on the left of the central vector of polynomials to be linear. Second, for a particular case, we determine an explicit system of invariants from which systems of equations whose solutions are the elements of an equivalence class can be derived.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Shor, P.W.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Comput. 26(5), 1484–1509 (1997)
Merkle, R.C.: Secrecy, authentication, and public key systems. PhD thesis, Stanford University (1979)
McEliece, R.J.: A public-key cryptosystem based on algebraic coding theory. Technical Report 42-44, Jet Propulsion Laboratory (1978)
Goldreich, O., Goldwasser, S., Halevi, S.: Public-key cryptosystems from lattice reduction problems. In: Kaliski Jr., B.S. (ed.) CRYPTO 1997. LNCS, vol. 1294, pp. 112–131. Springer, Heidelberg (1997)
Ding, J., Gower, J.E., Schmidt, D.: Multivariate Public Key Cryptosystems. Advances in Information Security, vol. 25. Springer (2006)
Garey, M.R., Johnson, D.S.: Computer and Intractability: A Guide to the Theory of NP-Completness. W. H. Freeman & Co. (1990)
Wolf, C.: Multivariate Quadratic Polynomials in Public Key Criptography. PhD thesis, Katholieke Universiteit Leuven (November 2005)
Patarin, J.: Cryptanalysis of the Matsumoto and Imai public key scheme of Eurocryptp’88. In: Coppersmith, D. (ed.) CRYPTO 1995. LNCS, vol. 963, pp. 248–261. Springer, Heidelberg (1995)
Feldmann, A.T.: A Survey of Attacks on Multivariate Cryptosystems. PhD thesis, University of Waterloo (2005)
Bouillaguet, C., Fouque, P.-A., Véber, A.: Graph-theoretic algorithms for the ‘isomorphism of polynomials’ problem. In: Johansson, T., Nguyen, P.Q. (eds.) EUROCRYPT 2013. LNCS, vol. 7881, pp. 211–227. Springer, Heidelberg (2013)
Matsumoto, T., Imai, H.: Public quadratic polynomial-tuples for efficient signature-verification and message-encryption. In: Günther, C.G. (ed.) EUROCRYPT 1988. LNCS, vol. 330, pp. 419–453. Springer, Heidelberg (1988)
Courtois, N., Klimov, A., Patarin, J., Shamir, A.: Efficient algorithms for solving overdefined systems of multivariate polynomial equations. In: Preneel, B. (ed.) EUROCRYPT 2000. LNCS, vol. 1807, pp. 392–407. Springer, Heidelberg (2000)
Courtois, N., Goubin, L., Meier, W., Tacier, J.-D.: Solving underdefined systems of multivariate quadratic equations. In: Naccache, D., Paillier, P. (eds.) PKC 2002. LNCS, vol. 2274, pp. 211–227. Springer, Heidelberg (2002)
Patarin, J.: Hidden fields equations (HFE) and isomorphisms of polynomials (IP): Two new families of asymmetric algorithms. In: Maurer, U.M. (ed.) EUROCRYPT 1996. LNCS, vol. 1070, pp. 33–48. Springer, Heidelberg (1996)
Patarin, J., Goubin, L., Courtois, N.: Improved algorithms for isomorphisms of polynomials. In: Nyberg, K. (ed.) EUROCRYPT 1998. LNCS, vol. 1403, pp. 184–200. Springer, Heidelberg (1998)
Wolf, C., Preneel, B.: Equivalent keys in multivariate quadratic public key systems. Journal of Mathematical Cryptology 4(4), 375–415 (2005)
Wolf, C., Preneel, B.: Large superfluous keys in \(\mathcal{M}\)ultivariate \(\mathcal{Q}\)uadratic asymmetric systems. In: Vaudenay, S. (ed.) PKC 2005. LNCS, vol. 3386, pp. 275–287. Springer, Heidelberg (2005)
Wolf, C., Preneel, B.: Equivalent keys in HFE, C*, and variations. In: Dawson, E., Vaudenay, S. (eds.) Mycrypt 2005. LNCS, vol. 3715, pp. 33–49. Springer, Heidelberg (2005)
Faugère, J.-C., Perret, L.: Polynomial equivalence problems: Algorithmic and theoretical aspects. In: Vaudenay, S. (ed.) EUROCRYPT 2006. LNCS, vol. 4004, pp. 30–47. Springer, Heidelberg (2006)
Lin, D., Faugère, J.-C., Perret, L., Wang, T.: On enumeration of polynomial equivalence classes and their application to MPKC. Finite Fields and Their Applications 18(2), 283–302 (2012)
Mingjie, L., Lidong, H., Xiaoyun, W.: On the equivalent keys in multivariate cryptosystems. Tsinghua Science & Technology 16, 225–232 (2011)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Pena, M.C., Díaz, R.D., Encinas, L.H., Masqué, J.M. (2014). The Isomorphism of Polynomials Problem Applied to Multivariate Quadratic Cryptography. In: Herrero, Á., et al. International Joint Conference SOCO’13-CISIS’13-ICEUTE’13. Advances in Intelligent Systems and Computing, vol 239. Springer, Cham. https://doi.org/10.1007/978-3-319-01854-6_58
Download citation
DOI: https://doi.org/10.1007/978-3-319-01854-6_58
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-01853-9
Online ISBN: 978-3-319-01854-6
eBook Packages: EngineeringEngineering (R0)