Small Solutions to Polynomial Equations, and Low Exponent RSA Vulnerabilities

We show how to find sufficiently small integer solutions to a polynomial in a single variable modulo N, and to a polynomial in two variables over the integers. The methods sometimes extend to more variables. As applications: RSA encryption with exponent 3 is vulnerable if the opponent knows two-thirds of the message, or if two messages agree over eight-ninths of their length; and we can find the factors of N=PQ if we are given the high order \(\frac{1}{4} \log_2 N\) bits of P.

Authors and Affiliations

1. IBM Research, T. J. Watson Research Center , Yorktown Heights, NY 10598, U.S.A., US

   Don Coppersmith

Received 21 December 1995 and revised 11 August 1996

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