Reduction to cdp,dl or cdh? Assuming a randomize encoding scheme that takes as input a secret key $\mathsf{sk} \in \mathbb{Z}_p$ for a large prime number $p$. Then the algorithm outputs $g^{\mathsf{sk}x}, x \in \mathbb{Z}_p^*$. Analyzing the privacy of the scheme where we can reduce its security for dinstinguishing encodings of $x$ based on an indistinguishability based definition?
Actually, the algorithm is distinguishable.

Let us assume that the attacker has two input, output pairs $(x, fx), (y, fy)$.  Then, he can test whether $fy^x = fx^y$.  If $fx = g^{sk\cdot x}$ and $fy = g^{sk\cdot y}$, this test will succeed, as $fy^x = g^{sk\cdot y \cdot x} = g^{sk\cdot x \cdot y} = fx^y$.

Furthermore, given $x$ and $g^{sk \cdot x}$, the attacker can (with nontrivial probability) recover $g^{sk}$.  The nontrivial probability is that whether $x^{-1} \bmod p-1$ exists; if it does, then $g^{sk} = (g^{sk \cdot x})^{x^{-1}}$.  The attacker can use that to evaluate the function for any input.

The above assumes that the attacker knows $p$.  Even if he doesn't, he can still practically distinguish (and recover $p$) if the sets he was given had inputs there were either


Small, for example, if the inputs $x, y, z$ are sufficiently small that the attacker can compute $\gcd(fy^x-fx^y, fz^x-fx^z)$
Related (e.g. $x, 2x, 4x$ with the corresponding outputs).