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If I have an encrypted value $Enc(x)$ with Paillier cryptosystem, is it feasible to compute an encryption form of $\ln(x)$ or its approximation using homomorphic properties? The input $x$ is always positive. The $\ln$ function satisfies $\ln(1)=0$ and the derivative of $\ln$ function is $1/x$. Or if I want to compute its approximation, what other information do I need?

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closed as unclear what you're asking by kelalaka, AleksanderRas, Maeher, Squeamish Ossifrage, Maarten Bodewes Sep 21 at 21:48

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ Yes, it is the $\ln$ with $1/x$ derivative and be $0$ when $x=1$. The input $x$ is guaranteed to be positive. I have edited the question. Do you have any suggestions? Thank you. $\endgroup$ – WYC Sep 18 at 6:34
  • $\begingroup$ I do not understand. Maybe I didn't express it clearly. What I desired is to compute an encryption form of $\ln(x)$ given $Enc(x)$ using Paillier cryptosystem. $\endgroup$ – WYC Sep 18 at 8:16
  • $\begingroup$ What space are you computing $\ln$ in? What does, e.g., $\ln x$ in $\mathbb Z/(p^2 q^2)\mathbb Z$ mean? What does the derivative of a (non-polynomial) function mean in this space? $\endgroup$ – Squeamish Ossifrage Sep 19 at 18:38
  • $\begingroup$ I am not familiar with Cryptographic, and I have no idea about the importance of the mean of derivative of a function in this space. I suppose $x$ is in the plaintext space, while $Enc(x)$ is in the ciphertext space, for example, I want someone else to evaluate a sigmoid function (or other non-polynomial functions) given $Enc(x)$. It seems that it is possible if convert the evaluation to secure multiparty computation, but I am not sure whether Paillier alone could do the approximate evaluation. Hope to receive any valuable input or suggestion from you. $\endgroup$ – WYC Sep 21 at 5:51