Homomorphic Encryption with Addition and Exponentiation Is there any homomorphic encryption scheme which supports addition and power over cipher text ? Paillier is close but it supports addition and multiplication with a constant.

I am getting an output like this:

1 0 1 0 -1 1


My goal is to make -1 positive by any means. As it will encrypted so I cannot know if it is -1 or 1. This (similar) output is being generated by subtracting binary streams.  

For example:

101101 -- A1
111001 --- A2
----------------------------
2 1 2 1 1 0 2 --- A+A2=A3
----------------------------
1 0 1 0 -1 1 ---- A3-1


A1 and A2 bits can be replaced to any other integers.
Also, can I perform AND operation in additive mode ?
Please note that this question is link to my previous question

Primary Objective:
To be able to check any of following


How many same bits occurs that is 1ns in A1 and A2 in the same position
How many zero bits on the same positions in A1 and A2
How many different bit locations in A1 and A2
(In most generality..) you're looking for fully homomorphic encryption (FHE), which supports homomorphic addition and homomorphic multiplication. To compute a plaintext's exponentiation, you would run the repeating squaring algorithm (using homomorphic multiplication), to generate an (evaluated) homomorphic ciphertext containing the input-plaintext's exponentiation.

Craig Gentry gave the breakthrough construction of FHE in 2009. Since then, there's been a titanic volume of work improving the techniques, and multiple surveys are available by searching online.

An open-source implementation of modern FHE -- the library HElib by Shai Halevi and Victor Shoup -- can be found here: https://github.com/shaih/HElib