Recover secret $x$ when $c\equiv m^x \pmod p$ with public $p$ (modified)

Given an encryption system where $c\equiv m^x \pmod p$, $p$ is a known prime,
1. Is it possible to recover $x$ with a known plaintext attack? Given $(p,\text{factorization of }\varphi(p),m,c)$
2. Is it possible to recover $x$ with a chosen plaintext attack? (Somehow with the chinese remainder theorem on the factorization of $\varphi(p)$ comes to mind)

• Yes fgrieu are right. – Meysam Ghahramani Jan 10 '16 at 15:04
• Hint: that's an (imprecise) statement of the Discrete Logarithm problem. Depending on parameters (including the domain for $x$, $p$, its size, the smoothness of $p-1$, the choice of $m$), that's hard, or feasible. – fgrieu Jan 10 '16 at 15:05
• Hint: you want to consider the Pohlig-Hellman algorithm. – fgrieu Jan 10 '16 at 16:24
• Note that $\varphi(p)$ very well can be $\varphi(p)=2q+1, q\in\mathbb P$ rendering this problem really hard. – SEJPM Jan 11 '16 at 20:32