Is it possible to encrypt a CNF? Is it possible to convert a CNF $\mathcal C$ into another CNF $\Psi(\mathcal C)$ such that


The function $\Psi$ can be computed in polynomial time from some secret random parameter $r$.
$\Psi(\mathcal C)$ has a solution if and only if $\mathcal C$ has a solution.
Any solution $x$ of $\Psi(\mathcal C)$ can be efficiently converted into a solution of $\mathcal C$ using $r$.
Without $r$, the solution $x$ (or any other property of $\Psi(\mathcal C)$) does not give any help in solving $\mathcal C$.


If there is such a $\Psi$, then it can be used to make others to solve computational challenges for us (with possibly replacing solving a CNF with other problems - I chose CNF because I wanted to make the problem more specific), in such a way that they cannot profit from a possible solution even if they know what problem we've used them to solve.
For example, we could embed a factorization problem into a computer game, which enables players to play only if they work on our problem in the background, from time to time sending back proofs of computation.
Maybe software can be even made "free" this way, where "free" hides a (possibly higher) cost in your parents' electricity bill.
The application you mention is called "proof of useful work" in the literature, see for instance this article.

You can use a fully homomorphic encryption scheme (where the plaintext is the CNF instance) to delegate the computation to an untrusted party without disclosing the input.

This doesn't answer exactly your question, since such scheme doesn't map a CNF into another CNF, but it does work for the intended application.