Advertisement

Finding Small Solutions to Small Degree Polynomials

  • Don Coppersmith
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2146)

Abstract

This talk is a brief survey of recent results and ideas concerning the problem of finding a small root of a univariate polynomial mod N, and the companion problem of finding a small solution to a bivariate equation over ℤ. We start with the lattice-based approach from [2,3], and speculate on directions for improvement.

Keywords

Modular polynomials lattice reduction 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Dan Boneh, personal communication.Google Scholar
  2. 2.
    D. Coppersmith, Finding a small root of a univariate modular equation. Advances in Cryptology-EUROCRYPT’96, LNCS 1070, Springer, 1996, 155–165.Google Scholar
  3. 3.
    D. Coppersmith, Finding a small root of a bivariate integer equation; factoring with high bits known, Advances in Cryptology-EUROCRYPT’96, LNCS 1070, Springer, 1996, 178–189.Google Scholar
  4. 4.
    D. Coppersmith, Small solutions to polynomial equations, and low exponent RSA vulnerabilities. J. Crypt. vol 10 no 4 (Autumn 1997), 233–260.zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    D. Coppersmith, N.A. Howgrave-Graham, S.V. Nagaraj, Divisors in Residue classes—Constructively. Manuscript.Google Scholar
  6. 6.
    N. Elkies, Rational points near curves and small nonzero |x 3-y2| via lattice reduction, ANTS-4, LNCS vol 1838 (2000) Springer Verlag, 33–63.Google Scholar
  7. 7.
    J. Håstad, On using RSA with low exponent in a public key network, Advances in Cryptology-CRYPTO’85, LNCS 218, Springer-Verlag, 1986, 403–408.Google Scholar
  8. 8.
    N.A. Howgrave-Graham, Finding small solutions of univariate modular equations revisited. Cryptography and Coding LNCS vol 1355. (1997) Springer-Verlag. 131–142.CrossRefGoogle Scholar
  9. 9.
    N.A. Howgrave-Graham, personal communication, 1997.Google Scholar
  10. 10.
    N.A. Howgrave-Graham, Approximate Integer Common Divisors, This volume, pp. 51–66.Google Scholar
  11. 11.
    C.S. Jutla, On finding small solutions of modular multivariate polynomial equations, Advances in Cryptology-EUROCRYPT’98, LNCS 1403, Springer, 1998, 158–170.CrossRefGoogle Scholar
  12. 12.
    S.V. Konyagin and T. Steger, On polynomial congruences, Mathematical Notes Vol 55 No 6 (1994), 596–600.MathSciNetCrossRefGoogle Scholar
  13. 13.
    A.K. Lenstra, H.W. Lenstra, and L. Lovasz, Factoring polynomials with rational coefficients, Math. Ann. 261 (1982), 515–534.zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    H. W. Lenstra, Jr., “Divisors in Residue Classes,” Mathematics of Computation, volume 42, number 165, January 1984, pages 331–340.zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    H.W. Lenstra, personal communication.Google Scholar
  16. 16.
    K.L. Manders and L.M. Adleman, NP-Complete Decision Problems for Binary Quadratics. JCSS 16(2), 1978, 168–184.zbMATHMathSciNetGoogle Scholar
  17. 17.
    Phong Nguyen, personal communication.Google Scholar
  18. 18.
    T.J. Rivlin, Chebyshev Polynomials, From Approximation Theory to Algebra and Number Theory, Wiley (1990).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Don Coppersmith
    • 1
  1. 1.IBM Research, T.J. Watson Research CenterYorktown HeightsUSA

Personalised recommendations