Finding a Small Root of a Univariate Modular Equation

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 less than about N(sup 1/k). We give an application to an RSA encryption protocol: 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.

By: Don Coppersmith

Published in: RC20223 in 1995


This Research Report is available. This report has been submitted for publication outside of IBM and will probably be copyrighted if accepted for publication. It has been issued as a Research Report for early dissemination of its contents. In view of the transfer of copyright to the outside publisher, its distribution outside of IBM prior to publication should be limited to peer communications and specific requests. After outside publication, requests should be filled only by reprints or legally obtained copies of the article (e.g., payment of royalties). I have read and understand this notice and am a member of the scientific community outside or inside of IBM seeking a single copy only.

Questions about this service can be mailed to .