Prime modulus for RSA and sharing a secret?
I think the confusion lies in mixing two distinct hard problems.

RSA is related to the difficulty of factoring large numbers. The scheme you link to in the paper is based on the difficulty of solving discrete logarithms. Since they are very different hard problems, you can't really compare the two in terms of what must be kept secret. 

For RSA, $\varphi(n)$ must be kept secret since with that you could factor $n$ (and hence make the hard problem easy). With the scheme you link to, you must keep the exponent secret as that is what you are trying to solve for when computing a discrete logarithm.