# Compute ln function of a Paillier encrypted value [closed]

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?

• 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. – WYC Sep 18 '19 at 6:34
• 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. – WYC Sep 18 '19 at 8:16
• 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? – Squeamish Ossifrage Sep 19 '19 at 18:38
• 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. – WYC Sep 21 '19 at 5:51