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International Conference on the Theory and Application of Cryptology and Information Security

ASIACRYPT 2006: Advances in Cryptology – ASIACRYPT 2006 pp 194–209Cite as

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On the Provable Security of an Efficient RSA-Based Pseudorandom Generator

On the Provable Security of an Efficient RSA-Based Pseudorandom Generator

  • Ron Steinfeld17,
  • Josef Pieprzyk17 &
  • Huaxiong Wang17 
  • Conference paper
  • 1671 Accesses

  • 19 Citations

Part of the Lecture Notes in Computer Science book series (LNSC,volume 4284)

Abstract

Pseudorandom Generators (PRGs) based on the RSA inversion (one-wayness) problem have been extensively studied in the literature over the last 25 years. These generators have the attractive feature of provable pseudorandomness security assuming the hardness of the RSA inversion problem. However, despite extensive study, the most efficient provably secure RSA-based generators output asymptotically only at most O(logn) bits per multiply modulo an RSA modulus of bitlength n, and hence are too slow to be used in many practical applications.

To bring theory closer to practice, we present a simple modification to the proof of security by Fischlin and Schnorr of an RSA-based PRG, which shows that one can obtain an RSA-based PRG which outputs Ω(n) bits per multiply and has provable pseudorandomness security assuming the hardness of a well-studied variant of the RSA inversion problem, where a constant fraction of the plaintext bits are given. Our result gives a positive answer to an open question posed by Gennaro (J. of Cryptology, 2005) regarding finding a PRG beating the rate O(logn) bits per multiply at the cost of a reasonable assumption on RSA inversion.

Keywords

  • Pseudorandom generator
  • RSA
  • provable security
  • lattice attack

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References

  1. Alexi, W., Chor, B., Goldreich, O., Schnorr, C.P.: RSA and Rabin Functions: Certain Parts Are as Hard as the Whole. SIAM Journal on Computing 17(2), 194–209 (1988)

    CrossRef  MATH  MathSciNet  Google Scholar 

  2. Ben-Or, M., Chor, B., Shamir, A.: On the Cryptographic Security of Single RSA Bits. In: Proc. 15-th STOC, pp. 421–430. ACM Press, New York (1983)

    Google Scholar 

  3. Berbain, C., Gilbert, H., Patarin, J.: QUAD: a Practical Stream Cipher with Provable Security. In: Vaudenay, S. (ed.) EUROCRYPT 2006. LNCS, vol. 4004, pp. 109–128. Springer, Heidelberg (2006)

    CrossRef  Google Scholar 

  4. Blackburn, S.R., Gomez-Perez, D., Gutierrez, J., Shparlinski, I.E.: Reconstructing Noisy Polynomial Evaluation in Residue Rings. Journal of Algorithms (to appear)

    Google Scholar 

  5. Blackburn, S.R., Gomez-Perez, D., Gutierrez, J., Shparlinski, I.E.: Predicting Nonlinear Pseudorandom Number Generators. Mathematics of Computation 74, 1471–1494 (2004)

    CrossRef  MathSciNet  Google Scholar 

  6. Blum, L., Blum, M., Shub, M.: A Simple Unpredictable Pseudo-Random Number Generator. SIAM Journal on Computing 15, 364–383 (1986)

    CrossRef  MATH  MathSciNet  Google Scholar 

  7. Blum, M., Micali, S.: How to Generate Cryptographically Strong Sequences of Pseudo-Random Bits. SIAM Journal on Computing 13, 850–864 (1984)

    CrossRef  MATH  MathSciNet  Google Scholar 

  8. Boneh, D., Durfee, G.: Cryptanalysis of RSA with private key d less than N 0.292. IEEE Trans. on Info. Theory 46(4), 1339–1349 (2000)

    CrossRef  MATH  MathSciNet  Google Scholar 

  9. Boneh, D., Halevi, S., Howgrave-Graham, N.A.: The Modular Inversion Hidden Number Problem. In: Boyd, C. (ed.) ASIACRYPT 2001. LNCS, vol. 2248, pp. 36–51. Springer, Heidelberg (2001)

    CrossRef  Google Scholar 

  10. Catalano, D., Gennaro, R., Howgrave-Graham, N., Nguyen, P.: Paillier’s Cryptosystem Revisited. In: Proc. CCS 2001, November 2001, ACM, New York (2001)

    Google Scholar 

  11. Coppersmith, D.: Small Solutions to Polynomial Equations, and Low Exponent RSA Vulnerabilities. J. of Cryptology 10, 233–260 (1997)

    CrossRef  MATH  MathSciNet  Google Scholar 

  12. Coppersmith, D.: Finding Small Solutions to Low Degree Polynomials. In: Silverman, J.H. (ed.) CaLC 2001. LNCS, vol. 2146, pp. 20–31. Springer, Heidelberg (2001)

    CrossRef  Google Scholar 

  13. Dai, W.: Crypto++ 5.2.1 Benchmarks (2006), http://www.eskimo.com/~weidai/benchmarks.html

  14. Fischlin, R., Schnorr, C.P.: Stronger Security Proofs for RSA and Rabin Bits. Journal of Cryptology 13, 221–244 (2000)

    CrossRef  MATH  MathSciNet  Google Scholar 

  15. Gennaro, R.: An Improved Pseudo-Random Generator Based on the Discrete-Logarithm Problem. Journal of Cryptology 18, 91–110 (2005)

    CrossRef  MATH  MathSciNet  Google Scholar 

  16. Goldreich, O.: Foundations of Cryptography, vol. I. Cambridge University Press, Cambridge (2003)

    Google Scholar 

  17. Goldreich, O., Rosen, V.: On the Security of Modular Exponentiation with Application to the Construction of Pseudorandom Generators. J. of Cryptology 16, 71–93 (2003)

    CrossRef  MATH  MathSciNet  Google Scholar 

  18. Goldwasser, S., Micali, S.: Probabilistic Encryption. J. of Computer and System Sciences 28(2), 270–299 (1984)

    CrossRef  MATH  MathSciNet  Google Scholar 

  19. Goldwasser, S., Micali, S., Tong, P.: Why and How to Establish a Private Code on a Public Network. In: Proc. FOCS 1982, pp. 134–144. IEEE Computer Society Press, Los Alamitos (1982)

    Google Scholar 

  20. Howgrave-Graham, N.: Finding Small Roots of Univariate Polynomials Revisited. In: Darnell, M.J. (ed.) Cryptography and Coding 1997. LNCS, vol. 1355, pp. 131–142. Springer, Heidelberg (1997)

    Google Scholar 

  21. Impagliazzo, R., Naor, M.: Efficient Cryptographic Schemes Provably as Secure as Subset Sum. Journal of Cryptology 9, 199–216 (1996)

    CrossRef  MATH  MathSciNet  Google Scholar 

  22. Lenstra, A.K., Lenstra, H.W., Lovász, L.: Factoring Polynomials with Rational Coefficients. Mathematische Annalen 261, 515–534 (1982)

    CrossRef  MATH  MathSciNet  Google Scholar 

  23. Lenstra, A.K., Verheul, E.R.: Selecting Cryptographic Key Sizes. J. of Cryptology 14, 255–293 (2001)

    MATH  MathSciNet  Google Scholar 

  24. Micali, S., Schnorr, C.P.: Efficient, Perfect Polynomial Random Number Generators. J. of Cryptology 3, 157–172 (1991)

    CrossRef  MATH  MathSciNet  Google Scholar 

  25. Nguyen, P.Q., Shparlinski, I.E.: The insecurity of the digital signature algorithm with partially known nonces. J. Cryptology 15, 151–176 (2002)

    CrossRef  MATH  MathSciNet  Google Scholar 

  26. Nguyen, P.Q., Stern, J.: The Two Faces of Lattices in Cryptology. In: Silverman, J.H. (ed.) CaLC 2001. LNCS, vol. 2146, pp. 146–180. Springer, Heidelberg (2001)

    CrossRef  Google Scholar 

  27. Patel, S., Sundaram, G.: An Efficient Discrete Log Pseudo Random Generator. In: Krawczyk, H. (ed.) CRYPTO 1998. LNCS, vol. 1462, pp. 304–317. Springer, Heidelberg (1998)

    Google Scholar 

  28. Sidorenko, A., Schoenmakers, B.: Concrete Security of the Blum-Blum-Shub Pseudorandom Generator. In: Smart, N.P. (ed.) Cryptography and Coding 2005. LNCS, vol. 3796, pp. 355–375. Springer, Heidelberg (2005)

    CrossRef  Google Scholar 

  29. Steinfeld, R., Pieprzyk, J., Wang, H.: On the Provable Security of an Efficient RSA-Based Pseudorandom Generator. Cryptology ePrint Archive, Report 2006/206 (2006), http://eprint.iacr.org/2006/206

  30. Vazirani, U.V., Vazirani, V.V.: Efficient and Secure Pseudo-Random Number Generation. In: Proc. FOCS 1984, pp. 458–463. IEEE Computer Society Press, Los Alamitos (1982)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Centre for Advanced Computing – Algorithms and Cryptography (ACAC), Dept. of Computing, Macquarie University, North Ryde, Australia

    Ron Steinfeld, Josef Pieprzyk & Huaxiong Wang

Authors
  1. Ron Steinfeld
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  2. Josef Pieprzyk
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  3. Huaxiong Wang
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Editor information

Editors and Affiliations

  1. Department of Computer Science and Engineering, Shanghai Jiao Tong University, 200240, Shanghai, China

    Xuejia Lai & Kefei Chen & 

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Steinfeld, R., Pieprzyk, J., Wang, H. (2006). On the Provable Security of an Efficient RSA-Based Pseudorandom Generator. In: Lai, X., Chen, K. (eds) Advances in Cryptology – ASIACRYPT 2006. ASIACRYPT 2006. Lecture Notes in Computer Science, vol 4284. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11935230_13

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  • DOI: https://doi.org/10.1007/11935230_13

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