Abstract
Recently, Wang et al proposed a new middle-field type scheme for multivariate public key encryption. There are three equations in the central map, so it is convenient to name it TH. They found that some linearization equations can be derived for TH and to overcome this defect, they combined the internal perturbation and plus methods to obtain an improved scheme which we call PTH+. They claimed that PTH+ can resist all known types of attacks, including differential attack, and to ensure it achieves a security level higher than 280, they suggested the parameter is taken as (l,r,m) = (47,6, 11). In this paper, we show that TH has a much weaker structure than what is analyzed by the inventors and it can be totally cracked by linearization attack. For PTH+, we propose a method to reduce the attack against PTH+ to an attack on TH+ (a plus variant of TH) using the property on its differentials, which was originally regarded as impossible by that authors. The total complexity of our attack is 2l + r + 1 (2l)w ≈ 272, which is independent on the number m of the additional random quadratic equations by the plus method and disproves the claim in their original paper that the larger is the m, the securer is PTH+.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Blake, I., Seroussi, G., Smart, N.: Elliptic Curves in Cryptography. Cambridge Unversity Press, Cambridge (1999)
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)
Diffie, W., Hellman, M.: New Directions in Cryptography. IEEE Transactions on Information Theory 22, 644–654 (1976)
Ding, J., Hu, L., Nie, X., Li, J., Wagner, J.: High Order Linearization Equation (HOLE) Attack on Multivariate Public Key Cryptosystems. In: Okamoto, T., Wang, X. (eds.) PKC 2007. LNCS, vol. 4450, pp. 233–248. Springer, Heidelberg (2007)
Ding, J.: A New Variant of the Matsumoto-Imai Through Perturbation. In: Bao, F., Deng, R., Zhou, J. (eds.) PKC 2004. LNCS, vol. 2947, pp. 305–318. Springer, Heidelberg (2004)
Ding, J., Gower, J.: Inoculating Multivariate Schemes Against Differential Attacks. In: Yung, M., Dodis, Y., Kiayias, A., Malkin, T.G. (eds.) PKC 2006. LNCS, vol. 3958, pp. 290–301. Springer, Heidelberg (2006)
Ding, J., Gower, J., Schmidt, D.: Multivariate Public-Key Cryptosystems. In: Advances in Information Security. Springer, Heidelberg (2006)
Ding, J., Schmidt, D.: Cryptanalysis of HEFV and the Internal Perturbation of HFE. In: Vaudenay, S. (ed.) PKC 2005. LNCS, vol. 3386, pp. 288–301. Springer, Heidelberg (2005)
Dinur, I., Shamir, A.: Cube Attacks on Tweakable Black Box Polynomials, http://eprint.iacr.org/2008/385
Faugère, J.: A New Efficient Algorithm for Computing Gröebner Bases (F4). Journal of Applied and Pure Algebra 139, 61–88 (1999)
Faugère, J.: A New Efficient Algorithm for Computing Gröebner Bases Without Reduction to Zero (F5). In: ISSAC, pp. 75–83. ACM Press, New York (2002)
Pierre-Alain, F., Granboulan, L., Stern, J.: Differential Cryptanalysis for Multivariate Schemes. In: Cramer, R. (ed.) EUROCRYPT 2005. LNCS, vol. 3494, pp. 341–353. Springer, Heidelberg (2005)
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 1988. In: Coppersmith, D. (ed.) CRYPTO 1995. LNCS, vol. 963, pp. 248–261. Springer, Heidelberg (1995)
Shor, P.: Polynomial-time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer. SIAM journal on computing 26, 1484–1509 (1997)
Wang, Z., Nie, X., Zheng, S., Yang, Y., Zhang, Z.: A New Construction of Multivariate Public Key Encryption Scheme through Internally Perturbed Plus. In: Gervasi, O., Murgante, B., Laganà, A., Taniar, D., Mun, Y., Gavrilova, M.L. (eds.) ICCSA 2008, Part I. LNCS, vol. 5072, pp. 1–13. Springer, Heidelberg (2008)
Wang, L., Yang, B., Hu, Y., Lai, F.: A Medium-Field Multivariate Public Key Encryption Scheme. In: Pointcheval, D. (ed.) CT-RSA 2006. LNCS, vol. 3860, pp. 132–149. Springer, Heidelberg (2006)
Yang, B., Chen, J.: All in the XL Family: Theory and Practice. In: Park, C.-s., Chee, S. (eds.) ICISC 2004. LNCS, vol. 3506, pp. 67–88. Springer, Heidelberg (2005)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Cao, W., Hu, L. (2009). Cryptanalysis of a Multivariate Public Key Encryption Scheme with Internal Perturbation Structure. In: Youm, H.Y., Yung, M. (eds) Information Security Applications. WISA 2009. Lecture Notes in Computer Science, vol 5932. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-10838-9_19
Download citation
DOI: https://doi.org/10.1007/978-3-642-10838-9_19
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-10837-2
Online ISBN: 978-3-642-10838-9
eBook Packages: Computer ScienceComputer Science (R0)