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I have a program implementing a function $f$, and want to find a pair of functions $e$ and $h$ such that $h \circ e = f$, but $e^{-1}$ cannot be recovered from $h$.

In other words, given some $e$-encrypted data, $h$ decrypts it and applies $f$. Knowing $h$ doesn't allow me to get back the unencrypted data, but rather only the result of $f$ applied to the unencrypted data.

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    $\begingroup$ What you're asking is unclear - when I try to interpret it, what I get from the first paragraph appears to be contradicted by what's in the second. Perhaps you could clarify your question? BTW: are you asking about 'data security' (given the output $f(x)$, it is hard to compute...' or 'program objuscation' (for example: given a description of $f$ and $h$ is hard to write the program that computes $e^{-1}$?) $\endgroup$
    – poncho
    Jun 30, 2023 at 22:09
  • $\begingroup$ Obfuscation. I basically want to combine the decryption function $e^{-1}$ and program $f$ in a way that the decryption function can't be separated from it. $\endgroup$
    – James
    Jun 30, 2023 at 22:14
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    $\begingroup$ The problem as you stated it is trivially solved by $h(x)=x, e(x)=f(x)$. Even if you want $e$ to be invertible with a key you can do $e_k(x) = (\text{Enc}_k(x), f(x))$ and $h((c, x)) = x$. $\endgroup$ Jul 1, 2023 at 2:48

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I believe you are describing Functional Encryption. I'm not very familiar with the literature, but the linked Wikipedia page and a search on ePrint might be good starting points.

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