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On Optimal Bounds of Small Inverse Problems and Approximate GCD Problems with Higher Degree

  • Noboru Kunihiro
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7483)

Abstract

We show a relation between optimal bounds of a small inverse problem and an approximate GCD problem. First, we present a lattice based method to solve small inverse problems with higher degree. The problem is a natural extension of small secret exponent attack on RSA cryptosystem introduced by Boneh and Durfee. They reduced this attack to solving a bivariate modular equation: \(x(A+y) \equiv 1 \pmod{e}\), where A is a given integer and e is a public exponent. They proved that the problem can be solved in polynomial time when d ≤ N 0.292. In this paper, we extend the Boneh–Durfee’s result to more general problem. For a monic polynomial h(y) of degree κ( ≥ 1), integers C and e, we want to find all small roots of a bivariate modular equation: \(xh(y)+C \equiv 0 \pmod{e}\). We denote by X and Y the upper bound of roots. We present an algorithm for solving the problem and prove that the problem can be solved in polynomial time if \(\gamma \leq 1-\sqrt{\kappa \alpha}\) and |C| is small enough, where X = e γ and Y = e α . We employ a similar approach as unravelled linearization technique introduced by Herrmann and May in especially evaluating the lattice volume. Interestingly, our algorithm does not rule out the case of C = 0, which implies that our algorithm can solve a univariate unknown modular equation \(h(y) \equiv 0 \pmod{p}\), where p is unknown. Our algorithm achieves the best bound in the literature. Then, we show that our obtained bound is natural under the similar sense of Howgrave-Graham’s discussion in CaLC2001 and we prove that our bound, including Boneh–Durfee’s bound, is optimal under the reasonable assumption.

Keywords

RSA Cryptosystem LLL algorithm Small Inverse Problem Approximate GCD Problem 

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References

  1. 1.
    Boneh, D., Durfee, G.: Cryptanalysis of RSA with private key d less than n 0.292. IEEE Transactions on Information Theory 46(4), 1339–1349 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Coppersmith, D.: Finding a Small Root of a Bivariate Integer Equation; Factoring with High Bits Known. In: Maurer, U.M. (ed.) EUROCRYPT 1996. LNCS, vol. 1070, pp. 178–189. Springer, Heidelberg (1996)Google Scholar
  3. 3.
    Coppersmith, D.: Finding a Small Root of a Univariate Modular Equation. In: Maurer, U.M. (ed.) EUROCRYPT 1996. LNCS, vol. 1070, pp. 155–165. Springer, Heidelberg (1996)Google Scholar
  4. 4.
    Coppersmith, D.: Small solutions to polynomial equations, and low exponent RSA vulnerabilities. J. Cryptology 10(4), 233–260 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Coron, J.-S., Mandal, A., Naccache, D., Tibouchi, M.: Fully Homomorphic Encryption over the Integers with Shorter Public Keys. In: Rogaway, P. (ed.) CRYPTO 2011. LNCS, vol. 6841, pp. 487–504. Springer, Heidelberg (2011)Google Scholar
  6. 6.
    van Dijk, M., Gentry, C., Halevi, S., Vaikuntanathan, V.: Fully Homomorphic Encryption over the Integers. In: Gilbert, H. (ed.) EUROCRYPT 2010. LNCS, vol. 6110, pp. 24–43. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  7. 7.
    Herrmann, M., May, A.: Attacking Power Generators Using Unravelled Linearization: When Do We Output Too Much? In: Matsui, M. (ed.) ASIACRYPT 2009. LNCS, vol. 5912, pp. 487–504. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  8. 8.
    Herrmann, M., May, A.: Maximizing Small Root Bounds by Linearization and Applications to Small Secret Exponent RSA. In: Nguyen, P.Q., Pointcheval, D. (eds.) PKC 2010. LNCS, vol. 6056, pp. 53–69. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  9. 9.
    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
  10. 10.
    Howgrave-Graham, N.: Approximate Integer Common Divisors. In: Silverman, J.H. (ed.) CaLC 2001. LNCS, vol. 2146, pp. 51–66. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  11. 11.
    Kunihiro, N.: Solving Generalized Small Inverse Problems. In: Steinfeld, R., Hawkes, P. (eds.) ACISP 2010. LNCS, vol. 6168, pp. 248–263. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  12. 12.
    Kunihiro, N.: Solving generalized small inverse problems. IEICE Transactions E94-A(6), 1274–1284 (2011)CrossRefGoogle Scholar
  13. 13.
    Kunihiro, N., Shinohara, N., Izu, T.: A Unified Framework for Small Secret Exponent Attack on RSA. In: Miri, A., Vaudenay, S. (eds.) SAC 2011. LNCS, vol. 7118, pp. 260–277. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  14. 14.
    Lenstra, A.K., Lenstra, H.W., Lovász, L.: Factoring polynomials with rational coefficients. Mathematische Annalen 261, 515–534 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    May, A.: New RSA Vulnerabilities Using Lattice Reduction Methods. Ph.D. thesis, University of Paderborn (2003)Google Scholar
  16. 16.
    Rivest, R.L., Shamir, A., Adleman, L.M.: A method for obtaining digital signatures and public-key cryptosystems. Commun. ACM 21(2), 120–126 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Wiener, M.J.: Cryptanalysis of short RSA secret exponents. IEEE Transactions on Information Theory 36(3), 553–558 (1990)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Noboru Kunihiro
    • 1
  1. 1.The University of TokyoJapan

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