Can we make subtraction result of cipher texts encrypted by Pascal Paillier absolute. Just like we use method Math.abs() in Java ? For example, if we subtract 0 from 1: 1-0 = 1, it is positive but 0-1 = -1. My requirement is that this subtraction should always be positive like:
1 - 0 = 1
0 - 1 = 1
Is there anyway i can achieve it please?
No, or at least, if you can, you have an Extremely Significant result; you've just shown that Paillier is a Fully Homomorphic system, and so it could perform any operation on encrypted data (and in a way that's significantly more efficient than any other known FHE system).
Here's why: The $|a - b|$ operation is effectively an $XOR$; if the ciphertexts $a, b$ are known to be either encrypted $0$ or $1$ values, then the result will be a 1 if they differ, and 0 if they are the same.
In addition with a bit more work, we can compute an $AND$. To do this, we could first compute $a + b - |a - b|$. This value will be an encrypted 2 if $a, b$ are both 1, and 0 if either is a 0. Then, multiplying this value by the constant $(n+1)/2$ (which can be done in $O(\log n)$ homomorphic additions) will give us an encrypted 1 if both $a, b$ are 1, 0 otherwise (or, on other words, an $AND$ operation).
The combination of the $AND$ and $XOR$ operations are complete; that is, any function (computable in bounded time) can be implemented by a sufficient number of them.
We don't believe that Paillier is an FHE system, hence either what you're asking for is infeasible, or we have a really exciting result on our hands.
