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?

  • $\begingroup$ 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 '19 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 '19 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 '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 '19 at 5:51