New Improved Probabilistic version of RSA
The paper is rather sloppily written, however it can be changed into the correct statement.

The statement in the paper is:


  She takes randomly elements $f \in F^*_p$ and checks whether or not $f^{2x}=1$. With probability at least $1 − 1/\ell$ she finds $f$ of order $\ell$.


As written, that's wrong.  However, if we modify it to:


  She takes a random element $f \in F^*_p$ and computes $g = f^{2x}$; if $g \ne 1$ (which will happen with probability $1 − 1/\ell$), then $g$ will be of order $\ell$.


Then it is a correct statement (and is the standard way of generating random elements of order $\ell$).

There are a number of places in the paper like this; places where, as written, it doesn't make much sense, but if you know your number theory, you can deduce what the author actually meant.  I suspect that this can be modified to make a secure cryptosystem; however whether it does have any advantage over RSA (and whether it's actually safe for multiple users to use the same modulus and different exponents) is far less clear.