Paper 2006/235

Application of ECM to a Class of RSA keys

Abderrahmane Nitaj

Abstract

Let $N=pq$ be an RSA modulus where $p$, $q$ are large primes of the same bitsize and $\phi(N)=(p-1)(q-1)$. We study the class of the public exponents $e$ for which there exist integers $X$, $Y$, $Z$ satisfying $$eX+\phi(N)Y=NZ,$$ with $\vert XY\vert <{\sqrt{2}\over 6}N^{1\over 2}$ and all prime factors of $\vert Y\vert$ are less than $10^{40}$. We show that these exponents are of improper use in RSA cryptosystems and that their number is at least $O\left(N^{{1\over 2}-\e}\right)$ where $\e$ is a small positive constant. Our method combines continued fractions, Coppersmith's lattice-based technique for finding small roots of bivariate polynomials and H. W. Lenstra's elliptic curve method (ECM) for factoring.