Advertisement

On the Instantiability of Hash-and-Sign RSA Signatures

  • Yevgeniy Dodis
  • Iftach Haitner
  • Aris Tentes
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7194)

Abstract

The hash-and-sign RSA signature is one of the most elegant and well known signatures schemes, extensively used in a wide variety of cryptographic applications. Unfortunately, the only existing analysis of this popular signature scheme is in the random oracle model, where the resulting idealized signature is known as the RSA Full Domain Hash signature scheme (RSA-FDH). In fact, prior work has shown several “uninstantiability” results for various abstractions of RSA-FDH, where the RSA function was replaced by a family of trapdoor random permutations, or the hash function instantiating the random oracle could not be keyed. These abstractions, however, do not allow the reduction and the hash function instantiation to use the algebraic properties of RSA function, such as the multiplicative group structure of ℤ n * . n. In contrast, the multiplicative property of the RSA function is critically used in many standard model analyses of various RSA-based schemes.

Motivated by closing this gap, we consider the setting where the RSA function representation is generic (i.e., black-box) but multiplicative, whereas the hash function itself is in the standard model, and can be keyed and exploit the multiplicative properties of the RSA function. This setting abstracts all known techniques for designing provably secure RSA-based signatures in the standard model, and aims to address the main limitations of prior uninstantiability results. Unfortunately, we show that it is still impossible to reduce the security of RSA-FDH to any natural assumption even in our model. Thus, our result suggests that in order to prove the security of a given instantiation of RSA-FDH, one should use a non-black box security proof, or use specific properties of the RSA group that are not captured by its multiplicative structure alone. We complement our negative result with a positive result, showing that the RSA-FDH signatures can be proven secure under the standard RSA assumption, provided that the number of signing queries is a-priori bounded.

Keywords

RSA Signature Hash-and-Sign Full Domain Hash Random Oracle Heuristic Generic Groups Black-Box Reductions 

References

  1. 1.
    Aggarwal, D., Maurer, U.: Breaking RSA Generically Is Equivalent to Factoring. In: Joux, A. (ed.) EUROCRYPT 2009. LNCS, vol. 5479, pp. 36–53. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  2. 2.
    Bellare, M., Boldyreva, A., Palacio, A.: An Uninstantiable Random-Oracle-Model Scheme for a Hybrid-Encryption Problem. In: Cachin, C., Camenisch, J. (eds.) EUROCRYPT 2004. LNCS, vol. 3027, pp. 171–188. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  3. 3.
    Bellare, M., Rogaway, P.: Random oracles are practical: A paradigm for designing efficient protocols. In: ACM Conference on Computer and Communications Security, pp. 62–73 (1993)Google Scholar
  4. 4.
    Bellare, M., Rogaway, P.: Optimal Asymmetric Encryption. In: De Santis, A. (ed.) EUROCRYPT 1994. LNCS, vol. 950, pp. 92–111. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  5. 5.
    Boldyreva, A., Fischlin, M.: Analysis of Random Oracle Instantiation Scenarios for OAEP and Other Practical Schemes. In: Shoup, V. (ed.) CRYPTO 2005. LNCS, vol. 3621, pp. 412–429. Springer, Heidelberg (2005)Google Scholar
  6. 6.
    Brown, J., González Nieto, J.M., Boyd, C.: Efficient CCA-Secure Public-Key Encryption Schemes from RSA-Related Assumptions. In: Barua, R., Lange, T. (eds.) INDOCRYPT 2006. LNCS, vol. 4329, pp. 176–190. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  7. 7.
    Canetti, R., Goldreich, O., Halevi, S.: On the Random-Oracle Methodology as Applied to Length-Restricted Signature Schemes. In: Naor, M. (ed.) TCC 2004. LNCS, vol. 2951, pp. 40–57. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  8. 8.
    Canetti, R., Goldreich, O., Halevi, S.: The random oracle methodology, revisited. JACM: Journal of the ACM, 51 (2004)Google Scholar
  9. 9.
    Coron, J.-S.: On the Exact Security of Full Domain Hash. In: Bellare, M. (ed.) CRYPTO 2000. LNCS, vol. 1880, pp. 229–235. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  10. 10.
    Cramer, R., Shoup, V.: Signature schemes based on the strong rsa assumption. ACM Trans. Inf. Syst. Secur. 3(3), 161–185 (2000)CrossRefGoogle Scholar
  11. 11.
    Dixon, J.D.: Asymptotically fast factorization of integers. Mathematics of Computation 36, 255–260 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Dodis, Y., Oliveira, R., Pietrzak, K.: On the Generic Insecurity of the Full Domain Hash. In: Shoup, V. (ed.) CRYPTO 2005. LNCS, vol. 3621, pp. 449–466. Springer, Heidelberg (2005)Google Scholar
  13. 13.
    Dodis, Y., Reyzin, L.: On the Power of Claw-Free Permutations. In: Cimato, S., Galdi, C., Persiano, G. (eds.) SCN 2002. LNCS, vol. 2576, pp. 55–73. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  14. 14.
    Dodis, Y., Haitner, I., Tentes, A.: On the instantiability of hash-and-sign rsa signatures. ePrint, http://eprint.iacr.org/2011/087
  15. 15.
    Gennaro, R., Gertner, Y., Katz, J., Trevisan, L.: Bounds on the efficiency of generic cryptographic constructions. SIAM Journal on Computing 35(1), 217–246 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    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
  17. 17.
    Gennaro, R., Trevisan, L.: Lower bounds on the efficiency of generic cryptographic constructions. In: Proceedings of the 41st Annual Symposium on Foundations of Computer Science, pp. 305–313. IEEE Computer Society (2000)Google Scholar
  18. 18.
    Goldwasser, S., Tauman-Kalai, Y.: On the (in)security of the fiat-shamir paradigm. In: Proceedings of the 44th Annual Symposium on Foundations of Computer Science (FOCS), pp. 102–113. IEEE Computer Society (2003)Google Scholar
  19. 19.
    Haitner, I., Hoch, J.J., Reingold, O., Segev, G.: Finding collisions in interactive protocols – A tight lower bound on the round complexity of statistically-hiding commitments. In: Proceedings of the 48th Annual Symposium on Foundations of Computer Science (FOCS), pp. 669–679. IEEE Computer Society (2007)Google Scholar
  20. 20.
    Haitner, I., Holenstein, T.: On the (Im)Possibility of Key Dependent Encryption. In: Reingold, O. (ed.) TCC 2009. LNCS, vol. 5444, pp. 202–219. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  21. 21.
    Hofheinz, D., Jager, T., Kiltz, E.: Short Signatures From Weaker Assumptions. In: Lee, D.H., Wang, X. (eds.) ASIACRYPT 2011. LNCS, vol. 7073, pp. 647–666. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  22. 22.
    Hohenberger, S., Waters, B.: Short and Stateless Signatures from the RSA Assumption. In: Halevi, S. (ed.) CRYPTO 2009. LNCS, vol. 5677, pp. 654–670. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  23. 23.
    Maurer, U.M.: Abstract models of computation in cryptography. In: IMA Int. Conf., pp. 1–12 (2005)Google Scholar
  24. 24.
    Miller, G.L.: Riemann’s hypothesis and tests for primality. Journal of Computer and System Sciences 13(3), 300–317 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Nechaev, V.I.: Complexity of a determinate algorithm for the discrete logarithm. MATHNASUSSR: Mathematical Notes of the Academy of Sciences of the USSR, 55 (1994)Google Scholar
  26. 26.
    Nielsen, J.B.: Separating Random Oracle Proofs from Complexity Theoretic Proofs: The Non-committing Encryption Case. In: Yung, M. (ed.) CRYPTO 2002. LNCS, vol. 2442, pp. 111–126. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  27. 27.
    Paillier, P.: Impossibility Proofs for RSA Signatures in the Standard Model. In: Abe, M. (ed.) CT-RSA 2007. LNCS, vol. 4377, pp. 31–48. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  28. 28.
    Paillier, P., Vergnaud, D.: Discrete-Log-Based Signatures May Not Be Equivalent to Discrete Log. In: Roy, B. (ed.) ASIACRYPT 2005. LNCS, vol. 3788, pp. 1–20. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  29. 29.
    Paillier, P., Villar, J.L.: Trading One-Wayness Against Chosen-Ciphertext Security in Factoring-Based Encryption. In: Lai, X., Chen, K. (eds.) ASIACRYPT 2006. LNCS, vol. 4284, pp. 252–266. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  30. 30.
    Pietrzak, K.: Compression from Collisions, or Why CRHF Combiners Have a Long Output. In: Wagner, D. (ed.) CRYPTO 2008. LNCS, vol. 5157, pp. 413–432. Springer, Heidelberg (2008)Google Scholar
  31. 31.
    Rivest, R.L., Shamir, A., Adelman, L.: A method for obtaining digital signatures and public-key cryptosystems. Communications of the ACM 21(2), 120–126 (1978)CrossRefzbMATHGoogle Scholar
  32. 32.
    RSA Laboratories, Redwood City, California. PKCS #1: RSA Encryption Standard (November 1993)Google Scholar
  33. 33.
    Shoup, V.: Lower Bounds for Discrete Logarithms and Related Problems. In: Fumy, W. (ed.) EUROCRYPT 1997. LNCS, vol. 1233, pp. 256–266. Springer, Heidelberg (1997)Google Scholar
  34. 34.
    Shoup, V.: Computational Introduction to Number Theory and Algebra. Cambridge University Press (2005)Google Scholar
  35. 35.
    Wee, H.: One-Way Permutations, Interactive Hashing and Statistically Hiding Commitments. In: Vadhan, S.P. (ed.) TCC 2007. LNCS, vol. 4392, pp. 419–433. Springer, Heidelberg (2007)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Yevgeniy Dodis
    • 1
  • Iftach Haitner
    • 2
  • Aris Tentes
    • 1
  1. 1.Department of Computer ScienceNew York UniversityUSA
  2. 2.School of Computer ScienceTel Aviv UniversityIsrael

Personalised recommendations