BMQE SYSTEM - A MQ Equations System based on Ergodic Matrix

Xiaoyi Zhou, Jixin Ma, Wencai Du, Bo Zhao, Miltos Petridis, Yongzhe Zhao

Abstract

In this paper, we propose a multivariate quadratic (MQ) equation system based on ergodic matrix (EM) over a finite field with q elements (denoted as F^q). The system actually implicates a problem which is equivalent to the famous Graph Coloring problem, and therefore is NP complete for attackers. The complexity of bisectional multivariate quadratic equation (BMQE) system is determined by the number of the variables, of the equations and of the elements of F^q, which is denoted as n, m, and q, respectively. The paper shows that, if the number of the equations is larger or equal to twice the number of the variables, and qn is large enough, the system is complicated enough to prevent attacks from most of the existing attacking schemes.

References

  1. Whitfield Diffie and Martin E. Hellman, 1976. “New directions in cryptography”. IEEE Transactions on Information Theory, Vol. IT-22 pp.644-654.
  2. Christopher Wolf, 2005. Multivariate Quadratic Polynomials in Public Key Cryptography. DIAMANT/EIDMA symposium 2005 on Technische Universiteit. [Online]. Available: http://www.win.tue. nl/diamant/ sym- posium05/abstracts/wolf.pdf.
  3. M. Herlihy and J. Wing, 1987. Axioms for Concurrent Objects. in Proc. the 14th ACM SIGACT-SIGPLAN symposium on Principles of programming languages. pp. 13-26.
  4. John Baena, Crystal Clough, Jintai Ding. 2008. “SquareVinegar Signature Scheme”, in Proc. PQCrypto 2008, pp. 17 - 30.
  5. Aviad Kipnis, Jacques Patarin, Louis Goubin, 1999. “Unbalanced oil and Vinegar Signature Schemes”, in Proc. EUROCRPT'99, pp. 206-222.
  6. Christopher Wolf, An Braeken, Bart Preneel, 2006. “On the Security of Stepwise Triangular Systems”. Designs Codes and Cryptography. Vol. 40(3): 285-302.
  7. Jacques Patarin. 1998. Cryptanalysis of the Matsumoto and Imai Public Key Scheme of Eurocypt'98. Codes and Cryptography. 20(2):175-209
  8. O. Hamdi, A. Bouallegue, S. Harari, 2006. “Hidden Field Equations Cryptosystem Performances”. in Proc. the IEEE International Conference on Computer Systems and Applications of AICCSA'06, pp.308-311.
  9. Jintai Ding, John Wagner, 2008. “Cryptanalysis of Rational Multivariate Public Key Cryptosystems”. in Proc. the 2nd International Workshop on PostQuantum Cryptography, pp. 124-136.
  10. Jacques Patarin, Louis Goubin, Nicolas T. Courtois, 1998. “C*-+ and HM: Variations Around Two Schemes of T. Matsumoto and H. Imai”. in Proc. the International Conference on the Theory and Application of Cryptology and Information Security ASIACRYPT 7898, pp. 35-49.
  11. Jintai Ding and Dieter Schmidt, 2006. Multivariate Public Key Cryptosystems, ser. Advances in Information Security, Berlin, Germany: Springer, vol. 25.: 288- 301
  12. Xijin Tang and Yong Feng, A new efficient algorithm for solving systems of multivariate polynomial equations, ser. Lecture Notes in Computer Science. Berlin, Germany: Springer, 2005, vol. 1807.
  13. Zhao Yongzhe, Wang Liou, Zhang Wei, 2004. “Information-Exchange Using the Ergodic Matrices in GF(2)”. in Proc. 2nd International Conference, ACNS 2004, pp. 388-397.
  14. Avid Arditti, Côme Berbain, Oliver Billet, Henri Gilbert, 2007. “Compact FPGA implementations of QUAD”. in Proc. the 2nd ACM symposium on Information, computer and communications security, pp. 347-349
  15. Aviad Kipnis, Adi Shamir. Cryptanalysis of the HFE Public Key Cryptosystem, 1999. in Proc. Advances in cryptology-CRYPTO 7899, 19th annual international cryptology conference. pp. 166-175.
  16. Arjen K. Lenstra, Eric R. Verheul, 2001. “Selecting Cryptographic Key Sizes”, J. Cryptology, Vol. 14(4), pp. 255-293.
  17. Aviad Kipnis, Adi Shamir, 1999. Cryptanalysis of the HFE Public Key Cryptosystem. in Proc. Advances in cryptology-CRYPTO 7899, 19th annual international cryptology conference. pp. 166-175.
  18. Christopher Wolf, An Braeken, Bart Preneel, 2006. “On the Security of Stepwise Triangular Systems”. Designs Codes and Cryptography. Vol. 40(3): 285-302.
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Paper Citation


in Harvard Style

Zhou X., Ma J., Du W., Zhao B., Petridis M. and Zhao Y. (2010). BMQE SYSTEM - A MQ Equations System based on Ergodic Matrix . In Proceedings of the International Conference on Security and Cryptography - Volume 1: SECRYPT, (ICETE 2010) ISBN 978-989-8425-18-8, pages 431-435. DOI: 10.5220/0002992304310435


in Bibtex Style

@conference{secrypt10,
author={Xiaoyi Zhou and Jixin Ma and Wencai Du and Bo Zhao and Miltos Petridis and Yongzhe Zhao},
title={BMQE SYSTEM - A MQ Equations System based on Ergodic Matrix},
booktitle={Proceedings of the International Conference on Security and Cryptography - Volume 1: SECRYPT, (ICETE 2010)},
year={2010},
pages={431-435},
publisher={SciTePress},
organization={INSTICC},
doi={10.5220/0002992304310435},
isbn={978-989-8425-18-8},
}


in EndNote Style

TY - CONF
JO - Proceedings of the International Conference on Security and Cryptography - Volume 1: SECRYPT, (ICETE 2010)
TI - BMQE SYSTEM - A MQ Equations System based on Ergodic Matrix
SN - 978-989-8425-18-8
AU - Zhou X.
AU - Ma J.
AU - Du W.
AU - Zhao B.
AU - Petridis M.
AU - Zhao Y.
PY - 2010
SP - 431
EP - 435
DO - 10.5220/0002992304310435