calculating a group element in a discrete log problem

An encryption scheme is based on the DLP: $c = a^x mod ~b$, where $x$ is hard to find. How hard is it to find $a$, given all the other values, $c$, $x$ and $b$?

Would this make the problem easy to solve?

$c^{1/x} = (a^x)^{1/x} mod ~b$

An example would be great to better understand.

• $c^{x^{-1}\pmod {\phi(b)}}=a \pmod b$ – Meysam Ghahramani Mar 17 '17 at 18:07

If you know the factorization of $b$, it's easy. One way would be to compute $x^{-1}$, the multiplicative inverse modulo $\phi(b)$, and then compute $c^{x^{-1}} \bmod b$
If you don't know the factorization of $b$, it's hard; in fact, it's essentially the RSA problem.
• why would i need to compute the multiplicative inverse modulo if it wont be used in the computation $c^{x^{-1}} mod ~b$ ? – user123 Mar 17 '17 at 18:12
• @user123: $c^{x^{-1}} \bmod b$ is meant as a sorthand for$$c^{(x^{-1}\bmod\phi(b)\;)} \bmod b$$ and thus makes use of $x^{-1}\bmod\phi(b)$, the multiplicative inverse of $x$ modulo $\phi(b)$. – fgrieu Mar 17 '17 at 18:27