Skip to main content

Finding a Small Root of a Univariate Modular Equation

Part of the Lecture Notes in Computer Science book series (LNCS,volume 1070)

Abstract

We show how to solve a polynomial equation (mod N) of degree k in a single variable x, as long as there is a solution smaller than N 1/k. We give two applications to RSA encryption with exponent 3. First, knowledge of all the ciphertext and 2/3 of the plaintext bits for a single message reveals that message. Second, if messages are padded with truly random padding and then encrypted with an exponent 3, then two encryptions of the same message (with different padding) will reveal the message, as long as the padding is less than 1/9 of the length of N. With several encryptions, another technique can (heuristically) tolerate padding up to about 1/6 of the length of N.

Keywords

  • Lattice Element
  • Short Vector
  • Small Exponent
  • Basis Reduction Method
  • Integer Linear Combination

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. D. Coppersmith, M. Franklin, J. Patarin and M. Reiter, “Low Exponent RSA with Related Messages,” Proceedings of Eurocrypt 96.

    Google Scholar 

  2. M. Franklin and M. Reiter, “A Linear Protocol Failure for RSA with Exponent Three,” presented at the rump session, Crypto 95, but not in the proceedings.

    Google Scholar 

  3. A. K. Lenstra, H. W. Lenstra and L. Lovasz, “Factoring Polynomials with Integer Coefficients,” Matematische Annalen 261 (1982), 513–534.

    MathSciNet  Google Scholar 

  4. B. Vallée, M. Girault and P. Toffin, “How to Guess -th Roots Modulo n by Reducing Lattice Bases,” Proceedings of AAECC-6, Springer LNCS 357 (1988) 427–442.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1996 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Coppersmith, D. (1996). Finding a Small Root of a Univariate Modular Equation. In: Maurer, U. (eds) Advances in Cryptology — EUROCRYPT ’96. EUROCRYPT 1996. Lecture Notes in Computer Science, vol 1070. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-68339-9_14

Download citation

  • DOI: https://doi.org/10.1007/3-540-68339-9_14

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-61186-8

  • Online ISBN: 978-3-540-68339-1

  • eBook Packages: Springer Book Archive