Proposal of a Signature Scheme Based on STS Trapdoor

  • Shigeo Tsujii
  • Masahito Gotaishi
  • Kohtaro Tadaki
  • Ryo Fujita
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6061)

Abstract

A New digital signature scheme based on Stepwise Triangular Scheme (STS) is proposed. The proposed trapdoor has resolved the vulnerability of STS and secure against both Gröbner Bases and Rank Attacks. In addition, as a basic trapdoor, it is more efficient than the existing systems. With the efficient implementation, the Multivariate Public Key Cryptosystems (MPKC) signature public key has the signature longer than the message by less than 25 %, for example.

Keywords

public key cryptosystem multivariate polynomial multivariate public key cryptosystem stepwise triangular scheme digital signature 

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References

  1. 1.
    Baena, J., Clough, C., Ding, J.: Square-Vinegar signature scheme. In: Buchmann, J., Ding, J. (eds.) PQCrypto 2008. LNCS, vol. 5299, pp. 17–30. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  2. 2.
    Braeken, A., Wolf, C., Preneel, B.: A study of the security of unbalanced oil and vinegar signature schemes. In: Menezes, A. (ed.) CT-RSA 2005. LNCS, vol. 3376, pp. 29–43. Springer, Heidelberg (2005)Google Scholar
  3. 3.
    Chen, J.M., Yang, B.Y.: A more secure and efficacious TTS signature scheme. In: Lim, J.-I., Lee, D.-H. (eds.) ICISC 2003. LNCS, vol. 2971, pp. 320–338. Springer, Heidelberg (2004)Google Scholar
  4. 4.
    Clough, C., Baena, J., Ding, J., Yang, B.Y., Chen, M.S.: Square, a new multivariate encryption scheme. In: Fischlin, M. (ed.) CT-RSA 2009. LNCS, vol. 5473, pp. 252–264. Springer, Heidelberg (2009)Google Scholar
  5. 5.
    Coppersmith, D., Stern, J., Vaudenay, S.: Attacks on the birational permutation signature schemes. In: Stinson, D.R. (ed.) CRYPTO 1993. LNCS, vol. 773, pp. 435–443. Springer, Heidelberg (1994)Google Scholar
  6. 6.
    Courtois, N., 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)CrossRefGoogle Scholar
  7. 7.
    Courtois, N., Goubin, L., Patarin, J.: SFLASHv3, a fast asymmetric signature scheme. Cryptology ePrint Archive, Report 2003/211 (October 2003), http://eprint.iacr.org/2003/211
  8. 8.
    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)Google Scholar
  9. 9.
    Ding, J., Schmidt, D.: Cryptanalysis of HFEv and internal perturbation of HFE. In: Vaudenay, S. (ed.) PKC 2005. LNCS, vol. 3386, pp. 288–301. Springer, Heidelberg (2005)Google Scholar
  10. 10.
    Ding, J., Schmidt, D.: Rainbow, a new multivariable polynomial signature scheme. In: Ioannidis, J., Keromytis, A.D., Yung, M. (eds.) ACNS 2005. LNCS, vol. 3531, pp. 164–175. Springer, Heidelberg (2005)Google Scholar
  11. 11.
    Ding, J., Wolf, C., Yang, B.Y.: ℓ-Invertible Cycles for \(\mathcal{M}\)ultivariate \(\mathcal{Q}\)uadratic (\(\mathcal{MQ}\)) public key cryptography. In: Okamoto, T., Wang, X. (eds.) PKC 2007. LNCS, vol. 4450, pp. 266–281. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  12. 12.
    Ding, J., Wagner, J.: Cryptanalysis of rational multivariate public key cryptosystems. In: Buchmann, J., Ding, J. (eds.) PQCrypto 2008. LNCS, vol. 5299, pp. 124–136. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  13. 13.
    Dubois, V., Granboulan, L., Stern, J.: Cryptanalysis of HFE with internal perturbation. In: Okamoto, T., Wang, X. (eds.) PKC 2007. LNCS, vol. 4450, pp. 249–265. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  14. 14.
    Dubois, V., Fouque, P.A., Shamir, A., Stern, J.: Practical cryptanalysis of SFLASH. In: Menezes, A. (ed.) CRYPTO 2007. LNCS, vol. 4622, pp. 1–12. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  15. 15.
    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)Google Scholar
  16. 16.
    Fouque, P.A., Granboulan, L., Stern, J.: Differential cryptanalysis for multivariate schemes. In: Cramer, R. (ed.) EUROCRYPT 2005. LNCS, vol. 3494, pp. 341–353. Springer, Heidelberg (2005)Google Scholar
  17. 17.
    Goubin, L., Courtois, N.: Cryptanalysis of the TTM cryptosystem. In: Okamoto, T. (ed.) ASIACRYPT 2000. LNCS, vol. 1976, pp. 44–57. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  18. 18.
    Hasegawa, S., Kaneko, T.: An attacking method for a public-key cryptosystem based on the difficulty of solving a system of non-linear equations. In: Proc. 10th SITA, JA5-3 (November 1987) (in Japanese)Google Scholar
  19. 19.
    Hashimoto, Y., Sakurai, K.: On construction of signature schemes based on birational permutations over noncommutative rings. In: Proceedings of the First International Conference on Symbolic Computation and Cryptography (SCC 2008), pp. 218–227 (2008)Google Scholar
  20. 20.
    Kasahara, M., Sakai, R.: A construction of public key cryptosystem for realizing ciphertext of size 100 bit and digital signature scheme. IEICE Transactions on Fundamentals E87-A(1) , 102–109 (2004)Google Scholar
  21. 21.
    Kasahara, M., Sakai, R.: A construction of public-key cryptosystem based on singular simultaneous equations. IEICE Transactions on Fundamentals E88-A(1), 74–80 (2005)CrossRefGoogle Scholar
  22. 22.
    Kipnis, A., Shamir, A.: Cryptanalysis of the oil and vinegar signature scheme. In: Krawczyk, H. (ed.) CRYPTO 1998. LNCS, vol. 1462, pp. 257–266. Springer, Heidelberg (1998)Google Scholar
  23. 23.
    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)Google Scholar
  24. 24.
    Matsumoto, T., Imai, H., Harashima, H., Miyakawa, H.: A class of asymmetric cryptosystems using obscure representations of enciphering functions. In: 1983 National Convention Record on Information Systems, IECE Japan, S8-5 (1983) (in Japanese)Google Scholar
  25. 25.
    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)Google Scholar
  26. 26.
    Moh, T.T.: A public key system with signature and master key functions. Communications in Algebra 27(5), 2207–2222 (1999)MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    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)Google Scholar
  28. 28.
    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)Google Scholar
  29. 29.
    Patarin, J.: The oil and vinegar signature scheme. Presented at the Dagstuhl Workshop on Cryptography (September 1997) (transparencies)Google Scholar
  30. 30.
    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)CrossRefGoogle Scholar
  31. 31.
    Patarin, J., Courtois, N., Goubin, L.: QUARTZ, 128-bit long digital signatures. In: Naccache, D. (ed.) CT-RSA 2001. LNCS, vol. 2020, pp. 282–297. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  32. 32.
    Shamir, A.: Efficient signature schemes based on birational permutations. In: Stinson, D.R. (ed.) CRYPTO 1993. LNCS, vol. 773, pp. 1–12. Springer, Heidelberg (1994)Google Scholar
  33. 33.
    Tsujii, S.: Public key cryptosystem using nonlinear equations. In: Proc. 8th SITA, December 1985, pp. 156–157 (1985) (in Japanese)Google Scholar
  34. 34.
    Tsujii, S., Kurosawa, K., Itoh, T., Fujioka, A., Matsumoto, T.: A public-key cryptosystem based on the difficulty of solving a system of non-linear equations. IEICE Transactions (D), J69-D(12) (1986) (1963–1970) (in Japanese)Google Scholar
  35. 35.
    Tsujii, S., Fujioka, A., Hirayama, Y.: Generalization of the public-key cryptosystem based on the difficulty of solving a system of non-linear equations. IEICE Transactions (A), J72-A(2), 390–397 (1989) (in Japanese); An English translation of [35] is included in [36] as an appendixGoogle Scholar
  36. 36.
    Tsujii, S., Tadaki, K., Fujita, R.: Piece in hand concept for enhancing the security of multivariate type public key cryptosystems: public key without containing all the information of secret key. Cryptology ePrint Archive, Report 2004/366 (December 2004), http://eprint.iacr.org/2004/366
  37. 37.
    Tsujii, S., Tadaki, K., Gotaishi, M., Fujita, R., Kasahara, M.: Proposal of PPS multivariate public key cryptosystems. Cryptology ePrint Archive, Report 2009/264 (June 2009), http://eprint.iacr.org/2009/264
  38. 38.
    Tsujii, S., Tadaki, K., Gotaishi, M., Fujita, R., Kasahara, M.: Proposal of integrated MPKC: PPS — STS enhanced by perturbed piece in hand method —. Technical Report of IEICE, ISEC2009-27, SITE2009-19, ICSS2009-41 (2009-2007) (July 2009) (in Japanese)Google Scholar
  39. 39.
    Wang, L.C., Chang, F.H.: Revision of tractable rational map cryptosystem. Cryptology ePrint Archive, Report 2004/046 (2006), http://eprint.iacr.org/2004/046
  40. 40.
    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)Google Scholar
  41. 41.
    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)CrossRefGoogle Scholar
  42. 42.
    Wolf, C., Braeken, A., Preneel, B.: Efficient cryptanalysis of RSE(2)PKC and RSSE(2)PKC. In: Blundo, C., Cimato, S. (eds.) SCN 2004. LNCS, vol. 3352, pp. 294–309. Springer, Heidelberg (2005)Google Scholar
  43. 43.
    Yang, B.Y., Chen, J.M.: 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)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Shigeo Tsujii
    • 1
  • Masahito Gotaishi
    • 1
  • Kohtaro Tadaki
    • 1
    • 2
  • Ryo Fujita
    • 1
  1. 1.Research & Development InitiativeChuo University 
  2. 2.JST CRESTTokyoJapan

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