3
$\begingroup$

So, in cryptography, it is sometimes the case that in order to set up a cryptosystem, one needs some output $f(x)$ for randomly generated $x$ where $f$ is a one-way function but where the input $x$ can be used to break the cryptosystem (i.e. the input $x$ is toxic waste that needs to be destroyed or the output $f(x)$ needs to be computed using some sort of secure multi-party computation).

Does there exist an efficient (by efficient I mean polynomial time) deterministic algorithm that given a one-way function $f$, returns a circuit $C_{f}$ such that for all inputs $x$, there is some $c$ where $C_{f}(x)=f(c)$ but where there is no efficient algorithm that given $x$ returns some $d$ with $f(d)=C_{f}(x)$ (preferably where the function $C_{f}$ is collision resistant)? Or does there exist some impossibility result for such a cryptosystem? If there is no impossibility result, then what is the weakest cryptosystem that ensures an algorithm $f\mapsto C_{f}$ exists?

$\endgroup$
  • $\begingroup$ There seems to be a trivial solution: pick a random $y$ in the image of $f$, and set $C_f$ to be the constant function that outputs $y$ on any input $x$. Then for any input $x$ there is $c$ such that $C_f(x)=y=f(c)$ as $y$ is in the image of $f$, but given any $x$ and $C_f$, finding $c$ is hard (it requires inverting $f$). $\endgroup$ – Geoffroy Couteau Apr 18 '18 at 21:40
  • $\begingroup$ @GeoffroyCouteau. I should have mentioned that the algorithm $H:f\mapsto C_{f}$ needs to be deterministic. $\endgroup$ – Joseph Van Name Apr 18 '18 at 23:02
  • $\begingroup$ @GeoffroyCouteau's solution can still work deterministically. E.g. if $f$ is a permutation you can choose the identity function $C_{f}(y) = y$, which is in the image of $f$. $\endgroup$ – pscholl Apr 19 '18 at 9:34
  • $\begingroup$ @GeoffroyCouteau: For general $f$, it's not feasible to find something in the range without evaluating $f(c)$ for some $c$ in the domain, and that $c$ is the "toxic waste." I think Joseph is asking whether it's possible to construct a securely oblivious computation which cryptographically randomly (using $x$ as the seed, in his notation) chooses an illegible representation of some $c$ from the domain, then returns $f(c)$, computed from that illegible representation. $\endgroup$ – Alex Coventry Jan 6 at 22:32

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.