My question is quite similar to Homomorphic modulo, but I want to give a context where the operation is carried in an outsourced environment.
Are there any specific homomorphic cryptographic schemes that could do the modular reduction and subtraction operations at the same time.
Suppose $Enc(x)$ a ciphertext of $x$ after homomorphic encryption, and $mod$ is modular reduction operation.
$$(Enc(x)-Enc(a))\ mod\ Enc(N) == (x-a)\ mod\ N$$ While $x$, $a$ and $N$ are all natural numbers. And $N$ are bigger than $x$ and $a$.
I know that $mod$ can be transformod into additional ($+$) and multiplicative ($*$) operations, which are operations supported by HE. But I don't want the factors exposed to someone who carry such operations, like $Server$.
$(6-5)\ mod\ 10 = 1+(0*10)=1$, $(4-5)\ mod\ 10 = -1+(1*10)=9$
But when a $Server$ is assigned do this, the factors like $0$ and $1$ here are exposed.
So I'm wondering, are there any specific HE schemes that can do modular reduction operation directly, or are there any solutions that can keep $Server$ from knowing the factors?