I'm implementing the Camenish-Lysyanskaya dynamic accumulator. It seems to me that the accumulator is provably secure because the trapdoor is unknown to the attacker.
The risks should be the fact that, given the Euler theorem, I can forge an element that is was not accumulated but still passes the test, namely:
$v^x\equiv v^{x\mod\phi}\mod n$
where $v$ is the accumulator without $x$, $\phi$ is the Euler function, and
$x\mod\phi\equiv x_f$
so we have a forged $x$ that passes the test.
What if I don't care about the factorization of n and I even publish $\phi$, but I just require that an acceptable integer $x_{acc}$ must be $A<x_{acc}<A+\phi$ with $A<\phi$.
This means that if, for example my input is the string "user1" this string will be hashed into an integer in the accepted interval.
If $\phi$ is large enough, I still can defy collisions and stop caring about keeping $\phi$ secret. All the possibly forged inputs fall automatically out of the specified interval so they will be discarded immediately. In other words there is no valid integer congruent to integer that maps the input "user1"
What do you think? Is this feasible? Or am I missing something obvious? I understand I may have skipped something important, I am not in the field, please ask for clarifications.