Is there an additive homomorphic encryption scheme which guarantees that if provided with
$E(v)$, $E(m_1)$ and $E(m_2)$
and $E(v)=E(m_1).E(m_2)$ then $v=m_1+m_2$
Please note this is not $v \equiv m_1+m_2 \pmod p$
To illustrate, assuming a group of order $p$, the additive version of ElGamal does not seem to match the criteria. Assuming $v=m_1+m_2$, there exists values $m'_1$ and $m'_2$ such that
$m'_1+m'_2 = m_1+m_2+k(p-1)$, $k \in Z^*$
which verifies $E(v)=E(m'_1)E(m'_2)$ due to Fermat's little theorem although $v$ is not equal to $m'_1+m'_2$
A simple numeric example is, assuming order $p=7$, a generator $g=2$, $v=4$, a shared secret $s$,
$m_1=1$ and $m_2=3$ obviously verifies that $E(1)E(3)=E(4) = 2^4.s \pmod 7 \equiv 2.s \pmod 7$
but so does $m_1=4$ and $m_2=6$ since $2^{10} \equiv 2 \pmod 7$