Secure Accumulators from Euclidean Rings without Trusted Setup

  • Helger Lipmaa
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7341)

Abstract

Cryptographic accumulators are well-known to be useful in many situations. However, the most efficient accumulator (the RSA accumulator) it is not secure against a certificate authority who has herself selected the RSA modulus n. We generalize previous work and define the root accumulator in modules over Euclidean rings. We prove that the root accumulator is secure under two different pairs of assumptions on the module family and on the used hash function. Finally, we propose a new instantiation of the root accumulator, based on class groups of imaginary quadratic order, that combines the best properties of previous solutions. It has short (non)membership proofs like the RSA accumulator, and at the same time it is secure against a malicious certificate authority. Up to this point, this seems to be the only unique application of class groups of imaginary quadratic orders, and we hope that this paper will motivate more research on cryptography in the said groups.

Keywords

Class groups of imaginary quadratic order cryptographic accumulators Euclidean rings 

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References

  1. 1.
    Barić, N., Pfitzmann, B.: Collision-Free Accumulators and Fail-Stop Signature Schemes without Trees. In: Fumy, W. (ed.) EUROCRYPT 1997. LNCS, vol. 1233, pp. 480–494. Springer, Heidelberg (1997)Google Scholar
  2. 2.
    Benaloh, J.C., de Mare, M.: One-Way Accumulators: A Decentralized Alternative to Digital Signatures. In: Helleseth, T. (ed.) EUROCRYPT 1993. LNCS, vol. 765, pp. 274–285. Springer, Heidelberg (1994)Google Scholar
  3. 3.
    Buchmann, J., Hamdy, S.: A Survey on IQ Cryptography. Technical Report TI-4/01, TU Darmstadt, Fachbereich Informatik (March 21, 2001)Google Scholar
  4. 4.
    Buchmann, J.A., Williams, H.C.: A Key-exchange System Based on Imaginary Quadratic Fields. Journal of Cryptology 1(2), 107–118 (1988)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Buldas, A., Laud, P., Lipmaa, H.: Accountable Certificate Management Using Undeniable Attestations. In: Jajodia, S., Samarati, P. (eds.) ACM CCS 2000, Athens, Greece, November 2-4, pp. 9–18. ACM Press (2000)Google Scholar
  6. 6.
    Buldas, A., Laud, P., Lipmaa, H.: Eliminating Counterevidence with Applications to Accountable Certificate Management. Journal of Computer Security 10(3), 273–296 (2002)Google Scholar
  7. 7.
    Buldas, A., Laud, P., Lipmaa, H., Villemson, J.: Time-Stamping with Binary Linking Schemes. In: Krawczyk, H. (ed.) CRYPTO 1998. LNCS, vol. 1462, pp. 486–501. Springer, Heidelberg (1998)Google Scholar
  8. 8.
    Buldas, A., Lipmaa, H., Schoenmakers, B.: Optimally Efficient Accountable Time-Stamping. In: Imai, H., Zheng, Y. (eds.) PKC 2000. LNCS, vol. 1751, pp. 293–305. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  9. 9.
    Camacho, P., Hevia, A., Kiwi, M., Opazo, R.: Strong Accumulators from Collision-Resistant Hashing. In: Wu, T.-C., Lei, C.-L., Rijmen, V., Lee, D.-T. (eds.) ISC 2008. LNCS, vol. 5222, pp. 471–486. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  10. 10.
    Camenisch, J., Kohlweiss, M., Soriente, C.: An Accumulator Based on Bilinear Maps and Efficient Revocation for Anonymous Credentials. In: Jarecki, S., Tsudik, G. (eds.) PKC 2009. LNCS, vol. 5443, pp. 481–500. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  11. 11.
    Camenisch, J., Lysyanskaya, A.: Dynamic Accumulators and Application to Efficient Revocation of Anonymous Credentials. In: Yung, M. (ed.) CRYPTO 2002. LNCS, vol. 2442, pp. 61–76. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  12. 12.
    Cohen, H.: A Course in Computational Algebraic Number Theory. Graduate Texts in Mathematics. Springer (1995)Google Scholar
  13. 13.
    Damgård, I., Fujisaki, E.: A Statistically-Hiding Integer Commitment Scheme Based on Groups with Hidden Order. In: Zheng, Y. (ed.) ASIACRYPT 2002. LNCS, vol. 2501, pp. 125–142. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  14. 14.
    Damgård, I., Koprowski, M.: Generic Lower Bounds for Root Extraction and Signature Schemes in General Groups. In: Knudsen, L.R. (ed.) EUROCRYPT 2002. LNCS, vol. 2332, pp. 256–271. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  15. 15.
    Gennaro, R., Halevi, S., Rabin, T.: Secure Hash-and-Sign Signatures without the Random Oracle. In: Stern, J. (ed.) EUROCRYPT 1999. LNCS, vol. 1592, pp. 123–139. Springer, Heidelberg (1999)Google Scholar
  16. 16.
    Groth, J., Sahai, A.: Efficient Non-interactive Proof Systems for Bilinear Groups. In: Smart, N.P. (ed.) EUROCRYPT 2008. LNCS, vol. 4965, pp. 415–432. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  17. 17.
    Haber, S., Stornetta, W.S.: How to Time-Stamp a Digital Document. In: Menezes, A., Vanstone, S.A. (eds.) CRYPTO 1990. LNCS, vol. 537, pp. 437–455. Springer, Heidelberg (1991)Google Scholar
  18. 18.
    Hamdy, S.: Computations in Class Groups of Imaginary Quadratic Number Fields. In: Innovations in Information Technology, Dubai, UAE, November 19-21, pp. 1–5 (2006)Google Scholar
  19. 19.
    Hamdy, S., Möller, B.: Security of Cryptosystems Based on Class Groups of Imaginary Quadratic Orders. In: Okamoto, T. (ed.) ASIACRYPT 2000. LNCS, vol. 1976, pp. 234–247. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  20. 20.
    Hühnlein, D., Takagi, T.: Reducing Logarithms in Totally Non-maximal Imaginary Quadratic Orders to Logarithms in Finite Fields. In: Lam, K.-Y., Okamoto, E., Xing, C. (eds.) ASIACRYPT 1999. LNCS, vol. 1716, pp. 219–231. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  21. 21.
    Jacobson Jr., M.J.: Subexponential Class Group Computation in Quadratic Orders. PhD thesis, Technische Universität Darmstadt, Fachbereich Informatik, Darmstadt, Germany (1999)Google Scholar
  22. 22.
    Lenstra, A.K., Lenstra, J. H.W. (eds.): The Development of the Number Field Sieve. Lecture Notes in Mathematics, vol. 1554. Springer, Heidelberg (1993)MATHGoogle Scholar
  23. 23.
    Li, J., Li, N., Xue, R.: Universal Accumulators with Efficient Nonmembership Proofs. In: Katz, J., Yung, M. (eds.) ACNS 2007. LNCS, vol. 4521, pp. 253–269. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  24. 24.
    Nguyen, L.: Accumulators from Bilinear Pairings and Applications. In: Menezes, A. (ed.) CT-RSA 2005. LNCS, vol. 3376, pp. 275–292. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  25. 25.
    Sander, T.: Efficient Accumulators without Trapdoor Extended Abstract. In: Varadharajan, V., Mu, Y. (eds.) ICICS 1999. LNCS, vol. 1726, pp. 252–262. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  26. 26.
    Sander, T., Ta-Shma, A., Yung, M.: Blind, Auditable Membership Proofs. In: Frankel, Y. (ed.) FC 2000. LNCS, vol. 1962, pp. 53–71. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  27. 27.
    Vollmer, U.: Asymptotically Fast Discrete Logarithms in Quadratic Number Fields. In: Bosma, W. (ed.) ANTS 2000. LNCS, vol. 1838, pp. 581–594. Springer, Heidelberg (2000)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Helger Lipmaa
    • 1
  1. 1.Institute of Computer ScienceUniversity of TartuEstonia

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