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Simple Matrix Scheme for Encryption

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Part of the Lecture Notes in Computer Science book series (LNSC,volume 7932)

Abstract

There are several attempts to build asymmetric pubic key encryption schemes based on multivariate polynomials of degree two over a finite field. However, most of them are insecure. The common defect in many of them comes from the fact that certain quadratic forms associated with their central maps have low rank, which makes them vulnerable to the MinRank attack. We propose a new simple and efficient multivariate pubic key encryption scheme based on matrix multiplication, which does not have such a low rank property. The new scheme will be called Simple Matrix Scheme or ABC in short. We also propose some parameters for practical and secure implementation.

Keywords

  • Multivariate Public Key Cryptosystem
  • Simple Matrix Scheme
  • MinRank Attack

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References

  1. Bosma, W., Cannon, J.J., Playoust, C.: The Magma algebra system I: the user language. J. Symb. Comput. 24(3-4), 235–265 (1997)

    MathSciNet  MATH  CrossRef  Google Scholar 

  2. Bettale, L., Faugère, J.-C., Perret, L.: Cryptanalysis of multivariate and odd-characteristic HFE variants. In: Catalano, D., Fazio, N., Gennaro, R., Nicolosi, A. (eds.) PKC 2011. LNCS, vol. 6571, pp. 441–458. Springer, Heidelberg (2011)

    CrossRef  Google Scholar 

  3. Goubin, L., Courtois, N.T.: Cryptanalysis of the TTM cryptosystem. In: Okamoto, T. (ed.) ASIACRYPT 2000. LNCS, vol. 1976, pp. 44–57. Springer, Heidelberg (2000)

    CrossRef  Google Scholar 

  4. Ding, J., Schmidt, D., Werner, F.: Algebraic attack on HFE revisited. In: Wu, T.-C., Lei, C.-L., Rijmen, V., Lee, D.-T. (eds.) ISC 2008. LNCS, vol. 5222, pp. 215–227. Springer, Heidelberg (2008)

    CrossRef  Google Scholar 

  5. Ding, J., Gower, J., Schmidt, D.: Multivariate Public Key Cryptography. Advances in Information Security series. Springer, Heidelberg (2006)

    Google Scholar 

  6. Ding, J., Yang, B.-Y., Chen, C.-H.O., Chen, M.-S., Cheng, C.-M.: New Differential-Algebraic Attacks and Reparametrization of Rainbow. In: Bellovin, S.M., Gennaro, R., Keromytis, A.D., Yung, M. (eds.) ACNS 2008. LNCS, vol. 5037, pp. 242–257. Springer, Heidelberg (2008)

    CrossRef  Google Scholar 

  7. 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)

    CrossRef  Google Scholar 

  8. Ding, J., Hodges, T.J.: Inverting HFE systems is quasi-polynomial for all fields. In: Rogaway, P. (ed.) CRYPTO 2011. LNCS, vol. 6841, pp. 724–742. Springer, Heidelberg (2011)

    CrossRef  Google Scholar 

  9. Faugère, J.C.: A new efficient algorithm for computing Gröbner bases (F4). J. Pure Appl. Algebra 139, 61–88 (1999)

    MathSciNet  MATH  CrossRef  Google Scholar 

  10. Faugère, J.-C., Levy-dit-Vehel, F., Perret, L.: Cryptanalysis of minRank. In: Wagner, D. (ed.) CRYPTO 2008. LNCS, vol. 5157, pp. 280–296. Springer, Heidelberg (2008)

    CrossRef  Google Scholar 

  11. Kipnis, A., Shamir, A.: Cryptanalysis of the Oil & Vinegar Signature Scheme. In: Krawczyk, H. (ed.) CRYPTO 1998. LNCS, vol. 1462, pp. 257–267. Springer, Heidelberg (1998)

    CrossRef  Google Scholar 

  12. Kipnis, A., Patarin, J., Goubin, L.: Unbalanced Oil and Vinegar Signature Schemes. In: Stern, J. (ed.) EUROCRYPT 1999. LNCS, vol. 1592, pp. 206–222. Springer, Heidelberg (1999)

    CrossRef  Google Scholar 

  13. 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)

    CrossRef  Google Scholar 

  14. Lidl, R., Niederreiter, H.: Finite Fields. Encyclopedia of Mathematics and its applications, vol. 20. Cambridge University Press

    Google Scholar 

  15. Moh, T.T.: A fast public key system with signature and master key functions. In: Proceedings of CrypTEC 1999, International Workshop on Cryptographic Techniques and E-Commerce, pp. 63–69. Hong-Kong City University Press (July 1999), http://www.usdsi.com/cryptec.ps

  16. 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)

    CrossRef  Google Scholar 

  17. Patarin, J.: The Oil and Vinegar Signature Scheme. Presented at the Dagstuhl Workshop on Cryptography (September 1997) (transparencies)

    Google Scholar 

  18. Patarin, J.: Cryptoanalysis 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)

    Google Scholar 

  19. 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)

    CrossRef  Google Scholar 

  20. Rivest, R., Shamir, A., Adleman, L.M.: A method for obtaining digital signatures and public-key cryptosystems. Communications of the ACM 21(2), 120–126

    Google Scholar 

  21. Shor, P.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM Journal on Computing 26(5), 1484–1509 (1997)

    MathSciNet  MATH  CrossRef  Google Scholar 

  22. 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)

    CrossRef  Google Scholar 

  23. Ding, J., Yang, B.-Y., Chen, C.-H.O., Chen, M.-S., Cheng, C.-M.: New differential-algebraic attacks and reparametrization of rainbow. In: Bellovin, S.M., Gennaro, R., Keromytis, A.D., Yung, M., et al. (eds.) ACNS 2008. LNCS, vol. 5037, pp. 242–257. Springer, Heidelberg (2008)

    CrossRef  Google Scholar 

  24. Thomae, E.: A Generalization of the Rainbow Band Separation Attack and its Applications to Multivariate Schemes. IACR Cryptology ePrint Archive (2012)

    Google Scholar 

  25. Buchmann, J.A., Ding, J., Mohamed, M.S.E., et al.: MutantXL: Solving multivariate polynomial equations for cryptanalysis. Symmetric Cryptography, 09031 (2009)

    Google Scholar 

  26. Mohamed, M.S.E., Mohamed, W.S.A.E., Ding, J., Buchmann, J.: MXL2: Solving polynomial equations over GF(2) using an improved mutant strategy. In: Buchmann, J., Ding, J., et al. (eds.) PQCrypto 2008. LNCS, vol. 5299, pp. 203–215. Springer, Heidelberg (2008)

    CrossRef  Google Scholar 

  27. Mohamed, M.S.E., Cabarcas, D., Ding, J., Buchmann, J., Bulygin, S.: MXL3: An efficient algorithm for computing gröbner bases of zero-dimensional ideals. In: Lee, D., Hong, S., et al. (eds.) ICISC 2009. LNCS, vol. 5984, pp. 87–100. Springer, Heidelberg (2010)

    CrossRef  Google Scholar 

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Tao, C., Diene, A., Tang, S., Ding, J. (2013). Simple Matrix Scheme for Encryption. In: Gaborit, P. (eds) Post-Quantum Cryptography. PQCrypto 2013. Lecture Notes in Computer Science, vol 7932. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38616-9_16

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  • DOI: https://doi.org/10.1007/978-3-642-38616-9_16

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-38615-2

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