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Is it possible to know the sign (positive or negative) of an homomorphic ciphertext particularly under paillier scheme ?

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Pallier encryption works over the finite ring modulo $n^2$. The distinction between positive and negative numbers don't exist there (or in any finite ring or field), so there isn't really a sign that works like the sign of integers.

Of course you can use signs in the notation, e.g. $-1$. But that is just the same element as $n^2-1$. Both represent the same residue class modulo $n^2$.

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  • $\begingroup$ Thanks ... so you mean that there is no way to know if the ciphertext represents a positive or negative plaintext ? $\endgroup$
    – witdev
    Oct 21, 2020 at 12:43
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    $\begingroup$ The plaintext is an element of the finite ring, too. It is not an integer. Integers are just used as representatives for the congruence classes. And the most common notation is the set $[0,1,...n^2-1]$. But especially with the finite structure in mind: they are not the integers themselves. In the integers the following would never work: $1+1+1+... = 0$. But it does work in finite rings. A bit more general: Finite rings are not ordered rings. The ordering does not exist, so you can't even do simple comparisons like $a \leq b$. $\endgroup$
    – tylo
    Oct 22, 2020 at 1:15
  • $\begingroup$ All non-bigmath computer integers are finite rings themselves. Just add 1 to [(unsigned char) 255] for example. $\endgroup$ Dec 18, 2021 at 1:14
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This paper purports to allow signed operations, as well as Arithmetic and logical operations https://www.sciencedirect.com/science/article/pii/S1319157815000804#b0035

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