Abstract
In the CT-track of the 2006 RSA conference, a new multivariate public key cryptosystem, which is called the Medium Field Equation (MFE) multivariate public key cryptosystem, is proposed by Wang, Yang, Hu and Lai. We use the second order linearization equation attack method by Patarin to break MFE. Given a ciphertext, we can derive the plaintext within 223 \(\mathbb {F}_{2^{16}}\)-multiplications, after performing once for any given public key a computation of complexity less than 252. We also propose a high order linearization equation (HOLE) attack on multivariate public key cryptosystems, which is a further generalization of the (first and second order) linearization equation (LE). This method can be used to attack extensions of the current MFE.
Chapter PDF
Similar content being viewed by others
Keywords
References
Akkar, M.-L., Courtois, N.T., Duteuil, R., Goubin, L.: A fast and secure implementation of Sflash. In: Desmedt, Y.G. (ed.) PKC 2003. LNCS, vol. 2567, pp. 267–278. Springer, Heidelberg (2002)
Ars, G., Faugère, J.-C., Imai, H., Kawazoe, M., Sugita, M.: Comparison between XL and Gröbner Basis Algorithms. In: Lee, P.J. (ed.) ASIACRYPT 2004. LNCS, vol. 3329, Springer, Heidelberg (2004)
Bardet, M., Fauge, J-C., Salvy, B., Yang, B-Y.: Asymptotic Behaviour of the Degree of Regularity of Semi-Regular Polynomial Systems. In: MEGA 2005, Eighth International Symposium on Effective Methods in Algebraic Geometry, Porto Conte, Alghero, Sardinia, Italy, May 27th - June 1st (2005)
Nicolas, T.: Courtois 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., 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., Pieprzyk, J.: Cryptanalysis of Block Ciphers with Overdefined Systems of Equations. In: Zheng, Y. (ed.) ASIACRYPT 2002. LNCS, vol. 2501, pp. 267–287. Springer, Heidelberg (2002)
Chen, J., Moh, T.: On the Goubin-Courtois attack on TTM. Cryptology ePrint Archive, 72 (2001), http://eprint.iacr.org/2001/072
Diffie, W., Hellman, M.: New directions in cryptography. IEEE Transactions on Information Theory 22(6), 644–654 (1976)
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, J., Gower, J.: Inoculating Multivariate Schemes Against Differential Attacks. Accepted for PKC-2006, IACR eprint 2005/255 (2005)
Ding, J., Schmidt, D.S.: A common defect of the TTM cryptosystem. In: Zhou, J., Yung, M., Han, Y. (eds.) ACNS 2003. LNCS, vol. 2846, pp. 68–78. Springer, Heidelberg (2003), http://eprint.iacr.org
Ding, J., Schmidt, D.S.: The new TTM implementation is not secure. In: Niederreiter, H., Feng, K.Q., Xing, C.P. (eds.) Proceedings of International Workshop on Coding, Cryptography and Combinatorics (CCC 2003), pp. 106–121 (2003)
Ding, J., Schmidt, D.S.: Cryptanalysis of HFEV and the internal perturbation of HFE. In: Vaudenay, S. (ed.) PKC 2005. LNCS, vol. 3386, pp. 288–301. Springer, Heidelberg (2005)
Ding, J., Schmidt, D.S.: Rainbow, a new multivariate public key signature scheme. In: Ioannidis, J., Keromytis, A.D., Yung, M. (eds.) ACNS 2005. LNCS, vol. 3531, pp. 164–175. Springer, Heidelberg (2005)
Faugère, J.-C.: A new efficient algorithm for computing Gröbner bases (F 4). Journal of Pure and Applied Algebra 139(1–3), 61–88 (1999)
Fouque, P.-A., Granboulan, L., Stern, J.: Differential Cryptanalysis for Multivariate Schemes. In: Cramer, R.J.F. (ed.) EUROCRYPT 2005. LNCS, vol. 3494, pp. 341–353. Springer, Heidelberg (2005)
Goubin, L., Courtois, N.: Cryptanalysis of the TTM cryptosystem. In: Okamoto, T. (ed.) ASIACRYPT 2000. LNCS, vol. 1976, pp. 44–57. Springer, Heidelberg (2000)
Kipnis, A., Shamir, A.: Cryptanalysis of the HFE public key cryptosystem by relinearization. In: Wiener, M.J. (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)
Moh, T.T.: A fast public key system with signature and master key functions. In: Lecture Notes at EE department of Stanford University (May 1999), http://www.usdsi.com/ttm.html
Moh, T., Chen, J.M., Yang, B.: Building Instances of TTM Immune to the Goubin-Courtois Attack and the Ding-Schmidt Attack. IACR eprint 2004/168 (2004), http://eprint.iacr.org
NESSIE. European project IST-1999-12324 on New European Schemes for Signature, Integrity and Encryption. http://www.cryptonessie.org
Patarin, J.: Cryptanalysis of the Matsumoto and Imai public key scheme of Eurocrypt’88. In: Coppersmith, D. (ed.) CRYPTO 1995. LNCS, vol. 963, pp. 248–261. Springer, Heidelberg (1995)
Patarin, J.: Hidden field equations (HFE) and isomorphism 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.: The oil and vinegar signature scheme. In: Dagstuhl Workshop on Cryptography, September 1997 (1997)
Patarin, J., Courtois, N., Goubin, L.: Flash, a fast multivariate signature algorithm. In: Naccache, D. (ed.) CT-RSA 2001. LNCS, vol. 2020, pp. 298–307. Springer, Heidelberg (2001)
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)
PQCrypto 2006: International Workshop on Post-Quantum Cryptography (2006), http://postquantum.cr.yp.to/
Rivest, R., Shamir, A., Adleman, L.M.: A method for obtaining digital signatures and public key cryptosystems. ACM 21(2), 120–126 (1978)
Shor, P.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM Rev. 41(2), 303–332 (1999)
Wang, L.-C., Hu, Y.-H., Lai, F., Chou, C.-Y., Yang, B.-Y.: Tractable Rational Map Signature. In: Vaudenay, S. (ed.) PKC 2005. LNCS, vol. 3386, pp. 244–257. Springer, Heidelberg (2005)
Wang, L.-C., Yang, B.-y., Hu, Y.-H., 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.: Building Secure Tame-like Multivariate Public key Cryptosystems–The New TTS. In: Boyd, C., González Nieto, J.M. (eds.) ACISP 2005. LNCS, vol. 3574, pp. 518–531. Springer, Heidelberg (2005)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer Berlin Heidelberg
About this paper
Cite this paper
Ding, J., Hu, L., Nie, X., Li, J., Wagner, J. (2007). High Order Linearization Equation (HOLE) Attack on Multivariate Public Key Cryptosystems. In: Okamoto, T., Wang, X. (eds) Public Key Cryptography – PKC 2007. PKC 2007. Lecture Notes in Computer Science, vol 4450. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-71677-8_16
Download citation
DOI: https://doi.org/10.1007/978-3-540-71677-8_16
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-71676-1
Online ISBN: 978-3-540-71677-8
eBook Packages: Computer ScienceComputer Science (R0)