A New Related Message Attack on RSA

  • Oded Yacobi
  • Yacov Yacobi
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3386)

Abstract

Coppersmith, Franklin, Patarin, and Reiter show that given two RSA cryptograms x e mod N and (ax+b) e mod N for known constants a,b ∈ ℤ N , one can compute x in O(elog 2 e) ℤ N -operations with some positive error probability. We show that given e cryptograms c i ≡ (a i x+b i ) e mod N, i=0,1,...e–1, for any known constants a i ,b i  ∈ ℤ N , one can deterministically compute x in O(e) ℤ N -operations that depend on the cryptograms, after a pre-processing that depends only on the constants. The complexity of the pre-processing is O(elog 2 e) ℤ N -operations, and can be amortized over many instances. We also consider a special case where the overall cost of the attack is O(e) ℤ N -operations. Our tools are borrowed from numerical-analysis and adapted to handle formal polynomials over finite-rings. To the best of our knowledge their use in cryptanalysis is novel.

References

  1. 1.
    Hopcroft, A., Ullman: The Design and Analysis of Computer Algorithms. Addison Wesley, Reading (1974); ISBN 0-201-00029-6Google Scholar
  2. 2.
    Boneh, D.: Twenty Years of Attacks on the RSA Cryptosystem. Notices of the American Mathematical Society (AMS) 46(2), 203–213 (1999)MATHMathSciNetGoogle Scholar
  3. 3.
    Bellare, M., Rogaway, P.: Optimal asymmetric encryption. In: Eurocrypt 1994, pp. 92–111 (1994)Google Scholar
  4. 4.
    Coppersmith, D., Franklin, M., Patarin, J., Reiter, M.: Low-exponent RSA with related messages. In: Maurer, U.M. (ed.) EUROCRYPT 1996. LNCS, vol. 1070, pp. 1–9. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  5. 5.
    Fujisaki, E., Okamoto, T., Pointcheval, D., Stern, J.: RSA-OAEP Is Secure Under the RSA Assumption. J. Crypt. 17(2), 81–104 (2004)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Rivest, R., Shamir, A., Adleman, L.M.: A Method for Obtaining Digital Signatures and Public-Key Cryptosystems. CACM 21(2), 120–126 (1978)MATHMathSciNetGoogle Scholar
  7. 7.
    Volkov, E.A.: Numerical Methods, p. 48. Hemisphere Publishing Corporation, New York (1987)MATHGoogle Scholar
  8. 8.
    Whittaker, E.T., Robinson: The Calculus of Observations: A Treatise on Numerical Mathematics, 4th edn., pp. 20–24. Dover, New York (1967)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Oded Yacobi
    • 1
  • Yacov Yacobi
    • 2
  1. 1.Department of MathematicsUniversity of California San DiegoLa JollaUSA
  2. 2.Microsoft ResearchOne Microsoft WayRedmondUSA

Personalised recommendations