Finding a Small Root of a Univariate Modular Equation

  • Don Coppersmith
Conference paper

DOI: 10.1007/3-540-68339-9_14

Part of the Lecture Notes in Computer Science book series (LNCS, volume 1070)
Cite this paper as:
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


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 N1/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.

Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • Don Coppersmith
    • 1
  1. 1.T.J. Watson Research CenterIBM ResearchYorktown HeightsUSA

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