Abstract
In this paper we propose a novel cryptanalytic method against multivariate schemes, which adapts differential cryptanalysis to this setting. In multivariate quadratic systems, the differential of the public key is a linear map and has invariants such as the dimension of the kernel. Using linear algebra, the study of this invariant can be used to gain information on the secret key. We successfully apply this new method to break the original Matsumoto-Imai cryptosystem using properties of the differential, thus providing an alternative attack against this scheme besides the attack devised by Patarin. Next, we present an attack against a randomised variant of the Matsumoto-Imai cryptosystem, called PMI. This scheme has recently been proposed by Ding, and according to the author, it resists all previously known attacks. We believe that differential cryptanalysis is a general and powerful method that can give additional insight on most multivariate schemes proposed so far.
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References
Bach, E., Shallit, J.: Algorithmic Number Theory. Efficient Algorithms, vol. 1. MIT Press, Cambridge (1996)
Bollobás, B.: Random Graphs, 2nd edn. Cambridge University Press, Cambridge (2001)
Cohen, H.: A Course in Computational Algebraic Number Theory. In: Graduate Texts in Mathematics, vol. 138. Springer, Heidelberg (1993)
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.T.: The security of Hidden Field Equations (HFE). In: Naccache, D. (ed.) CT-RSA 2001. LNCS, vol. 2020, pp. 266–281. Springer, Heidelberg (2001)
Courtois, N.T., Daum, M., Felke, P.: On the security of HFE, HFEv- and Quartz. In: Desmedt, Y.G. (ed.) PKC 2003. LNCS, vol. 2567, pp. 337–350. Springer, Heidelberg (2003), http://eprint.iacr.org/2002/138/
Ding, J.: A New Variant of the Matsumoto-Imai Cryptosystem through Perturbation. In: Bao, F., Deng, R., Zhou, J. (eds.) PKC 2004. LNCS, vol. 2947, pp. 305–318. Springer, Heidelberg (2004)
Ding-Feng, Y., Kwok-Yan, L., Zong-Duo, D.: Cryptanalysis of 2R schemes. In: Wiener, M. (ed.) CRYPTO 1999. LNCS, vol. 1666, pp. 315–325. Springer, Heidelberg (1999)
Faugère, J.-C., Joux, A.: Algebraic Cryptanalysis of Hidden Field Equation (HFE) Cryptosystems Using Gröbner Bases. In: Boneh, D. (ed.) CRYPTO 2003. LNCS, vol. 2729, pp. 44–60. Springer, Heidelberg (2003)
Gilbert, H., Minier, M.: Cryptanalysis of SFLASH. In: Knudsen, L.R. (ed.) EUROCRYPT 2002. LNCS, vol. 2332, pp. 288–298. Springer, Heidelberg (2002)
Kipnis, A., Shamir, A.: Cryptanalysis of the HFE Public Key Cryptosystem by Relinearization. In: Wiener, M. (ed.) CRYPTO 1999. LNCS, vol. 1666, pp. 19–30. Springer, Heidelberg (1999)
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)
Patarin, J.: Cryptanalysis of the Matsumoto and Imai Public Key Scheme of Eurocrypt 1998. In: Coppersmith, D. (ed.) CRYPTO 1995. LNCS, vol. 963, pp. 248–261. Springer, Heidelberg (1995)
Patarin, J.: Assymetric Cryptography with a Hidden Monomial. In: Koblitz, N. (ed.) CRYPTO 1996. LNCS, vol. 1109, pp. 45–60. Springer, Heidelberg (1996)
Patarin, J.: Hidden Fields Equations (HFE) and Isomorphisms of Polynomial (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.: \(C^{*}_{-+}\) and HM: Variations around Two Schemes of T.Matsumoto and H.Imai. In: Ohta, K., Pei, D. (eds.) ASIACRYPT 1998. LNCS, vol. 1514, pp. 35–50. Springer, Heidelberg (1998)
Patarin, J.: Hidden field 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)
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Fouque, PA., Granboulan, L., Stern, J. (2005). Differential Cryptanalysis for Multivariate Schemes. In: Cramer, R. (eds) Advances in Cryptology – EUROCRYPT 2005. EUROCRYPT 2005. Lecture Notes in Computer Science, vol 3494. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11426639_20
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DOI: https://doi.org/10.1007/11426639_20
Publisher Name: Springer, Berlin, Heidelberg
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