**RSA and a higher degree diophantine equation**

*Abderrahmane Nitaj*

**Abstract: **Let $N=pq$ be an RSA modulus where $p$, $q$ are large primes of the same bitsize. We study the class of the public exponents $e$ for which there exist an integer $m$ with $1\leq m\leq {\log{N}\over \log{32}}$ and small integers $u$, $X$, $Y$ and $Z$ satisfying $$(e+u)Y^m-\psi(N)X^m=Z,$$ where $\psi(N)=(p+1)(q-1)$. First we show that these exponents are of improper use in RSA cryptosystems. Next we show that their number is at least $O\left(mN^{{1\over 2}+{\a\over m}-\a-\e}\right)$ where $\a$ is defined by $N^{1-\a}=\psi(N)$.

**Category / Keywords: **public-key cryptography / RSA cryptosystem, Continued fractions, Coppersmith's algorithm

**Date: **received 9 Mar 2006

**Contact author: **nitaj at math unicaen fr

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**Version: **20060309:151057 (All versions of this report)

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