Reduction in Lossiness of RSA Trapdoor Permutation

  • Santanu Sarkar
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7644)

Abstract

We consider the lossiness of RSA trapdoor permutation studied by Kiltz, O’Neill and Smith in Crypto 2010. In Africacrypt 2011, Herrmann improved the cryptanalytic results of Kiltz et al. In this paper, we improve the bound provided by Herrmann, considering the fact that the unknown variables in the central modular equation of the problem are not balanced. We provide detailed experimental results to justify our claim. It is interesting that in many situations, our experimental results are better than our theoretical predictions. Our idea also extends the weak encryption exponents proposed by Nitaj in Africacrypt 2012.

Keywords

Multi-Prime Φ-Hiding Problem Lattice Modular Equation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Coppersmith, D.: Small Solutions to Polynomial Equations and Low Exponent Vulnerabilities. Journal of Cryptology 10(4), 223–260 (1997)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Fujioka, A., Okamoto, T., Miyaguchi, S.: ESIGN: An Efficient Digital Signature Implementation for Smart Cards. In: Davies, D.W. (ed.) EUROCRYPT 1991. LNCS, vol. 547, pp. 446–457. Springer, Heidelberg (1991)CrossRefGoogle Scholar
  3. 3.
    Herrmann, M., May, A.: Solving Linear Equations Modulo Divisors: On Factoring Given Any Bits. In: Pieprzyk, J. (ed.) ASIACRYPT 2008. LNCS, vol. 5350, pp. 406–424. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  4. 4.
    Herrmann, M.: Improved Cryptanalysis of the Multi-Prime φ - Hiding Assumption. In: Nitaj, A., Pointcheval, D. (eds.) AFRICACRYPT 2011. LNCS, vol. 6737, pp. 92–99. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  5. 5.
    Howgrave-Graham, N.: Finding Small Roots of Univariate Modular Equations Revisited. In: Darnell, M.J. (ed.) Cryptography and Coding 1997. LNCS, vol. 1355, pp. 131–142. Springer, Heidelberg (1997)Google Scholar
  6. 6.
    Jochemsz, E., May, A.: A Polynomial Time Attack on RSA with Private CRT-Exponents Smaller Than N0.073. In: Menezes, A. (ed.) CRYPTO 2007. LNCS, vol. 4622, pp. 395–411. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  7. 7.
    Jochemsz, E., May, A.: A Strategy for Finding Roots of Multivariate Polynomials with New Applications in Attacking RSA Variants. In: Lai, X., Chen, K. (eds.) ASIACRYPT 2006. LNCS, vol. 4284, pp. 267–282. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  8. 8.
    Kiltz, E., O’Neill, A., Smith, A.: Instantiability of RSA-OAEP under Chosen-Plaintext Attack. In: Rabin, T. (ed.) CRYPTO 2010. LNCS, vol. 6223, pp. 295–313. Springer, Heidelberg (2010), http://eprint.iacr.org/2011/559CrossRefGoogle Scholar
  9. 9.
    Lenstra, A.K., Lenstra Jr., H.W., Lovász, L.: Factoring polynomials with rational coefficients. Mathematische Annalen 261, 515–534 (1982)MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Lenstra Jr., H.W.: Factoring integers with elliptic curves. Annals of Mathematics 126, 649–673 (1987)MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    May, A.: Secret Exponent Attacks on RSA-type Schemes with Moduli N = prq. In: Bao, F., Deng, R., Zhou, J. (eds.) PKC 2004. LNCS, vol. 2947, pp. 218–230. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  12. 12.
    Nitaj, A.: A New Attack on RSA and CRT-RSA. In: Mitrokotsa, A., Vaudenay, S. (eds.) AFRICACRYPT 2012. LNCS, vol. 7374, pp. 221–233. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  13. 13.
    Rivest, R.L., Shamir, A., Adleman, L.: A Method for Obtaining Digital Signatures and Public Key Cryptosystems. Communications of ACM 21(2), 158–164 (1978)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Schridde, C., Freisleben, B.: On the Validity of the Φ-Hiding Assumption in Cryptographic Protocols. In: Pieprzyk, J. (ed.) ASIACRYPT 2008. LNCS, vol. 5350, pp. 344–354. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  15. 15.
    Takagi, T.: Fast RSA-type Cryptosystem Modulo pkq. In: Krawczyk, H. (ed.) CRYPTO 1998. LNCS, vol. 1462, pp. 318–326. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  16. 16.
    Tosu, K., Kunihiro, N.: Optimal Bounds for Multi-Prime Φ-Hiding Assumption. In: Susilo, W., Mu, Y., Seberry, J. (eds.) ACISP 2012. LNCS, vol. 7372, pp. 1–14. Springer, Heidelberg (2012)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Santanu Sarkar
    • 1
  1. 1.Indian Statistical InstituteKolkataIndia

Personalised recommendations