  The basic properties of RSA cryptosystems and some classical attacks on them
are described. Derived from geometric properties of the Euler functions, the
Euler function rays, a new ansatz to attack RSA cryptosystems is presented. A
resulting, albeit inefficient, algorithm is given. It essentially consists of a
loop with starting value determined by the Euler function ray and with step
width given by a function $\omega_e(n)$ being a multiple of the order
$\mathrm{ord}_n(e)$, where $e$ denotes the public key exponent and $n$ the RSA
modulus. For $n=pq$ and an estimate $r<\sqrt{pq}$ for the smaller prime factor
$p$, the running time is given by $T(e,n,r) = O((r-p)\ln e \ln n \ln r).$
