Is there an additively homomorphic encryption scheme that supports calculating a square root on the ciphertext? I need an additively homomorphic encryption scheme that satisfies: $D(\sqrt{E(m)}) \approx  \sqrt{m}$. It seems that the lifted ElGamal satisfies this, but it is hard to do decryption if the message space is large. Is there any other scheme that satisfies this?
I believe that such a system (with Homomorphic addition and approximate squareroots) is likely to be Fully Homomorphic, and hence it is unlikely that you could find an efficient candidate among existing systems (or, if you did, well, that's a Really Significant result).

Let us assume that:


Subtraction (as well as addition) can be done homomorphically
The approximate squareroot of 2 is 1.


Then, we can compute the "or" of two encrypted values $a$ and $b$ (which are both encrypted versions of either 0 or 1), by computing:

$$\sqrt{\sqrt{a+a+b+b}}$$

It is easy to verify that the result will be an encrypted 0 if both $a$ and $b$ are 0, and 1 if either (or both) are 1.

That, with homomorphic subtraction, is a complete set, and so we have a FHE system on our hands.