Solving Simultaneous Modular Equations of Low Degree

Abstract

We consider the problem of solving systems of equations $P_i (x) \equiv 0(\bmod n_i )i = 1 \cdots k$ where $P_i $ are polynomials of degree d and the $n_i $ are distinct relatively prime numbers and $x < \min (n_i )$. We prove that if $k > {{d(d + 1)} / 2}$ we can recover x in polynomial time provided $\min (n_i ) > 2^{d^2 } $. As a consequence the RSA cryptosystem used with a small exponent is not a good choice to use as a public-key cryptosystem in a large network. We also show that a protocol by Broder and Dolev [Proceedings on the 25th Annual IEEE Symposium on the Foundations of Computer Science, 1984] is insecure if RSA with a small exponent is used.