I am doing a little cryptography research and stuck with question.

Suppose $\bar m$ is a vector of 64-bit numbers. And i want to have an additive homomorphic encryption over them. I choose large (2048 bit) primes $n$, $s$ and vector $\bar r$ of large random numbers, with size of vector equal to vector $\bar m$. Finaly, encryption function $Enc$ looks like that: $$Enc(m_i) = (r_i,m_i+r_i\cdot s\mod n)$$ where $Enc$ yields an ordered pair of numbers. So the number $n$ with encrypted vector $c=Enc(m)$ goes to the opponent, but $s$ keeps secret.

Sum of an encrypted values is just sum of pairs (all operations are modulo $n$): $$Sum(c_i,c_j) = (c_i.first+c_j.first,c_i.second+c_j.second)$$ $$or$$ $$(a,b)+(c,d)=(a+c,b+d)$$ where $c_i.first$ is the first element of pair $c_i$ and $c_i.second$ is the second.

Decryption looks like that: $$Dec(c_i) = c_i.second - c_i.first\cdot s$$

Intuitively, i think, that this scheme is vulnerable, but i can't see why. I will be glad to have a hint to understanding this problem.

Many thanks in advance to your great community!
(sorry for any grammar mistakes, English is not my native language)

I am doing a little cryptography research and stuck with question.

Suppose $\bar m$ is a vector of 64-bit numbers. And i want to have an additive homomorphic encryption over them. I choose large (2048 bit) prime $n$, large not prime $s$ and vector $\bar r$ of large random numbers, with size of vector equal to vector $\bar m$. Finaly, encryption function $Enc$ looks like that: $$Enc(m_i) = (r_i,m_i+r_i\cdot s\mod n)$$ where $Enc$ yields an ordered pair of numbers. So the number $n$ with encrypted vector $c=Enc(m)$ goes to the opponent, but $s$ keeps secret.

Sum of an encrypted values is just sum of pairs (all operations are modulo $n$): $$Sum(c_i,c_j) = (c_i.first+c_j.first,c_i.second+c_j.second)$$ $$or$$ $$(a,b)+(c,d)=(a+c,b+d)$$ where $c_i.first$ is the first element of pair $c_i$ and $c_i.second$ is the second.

Decryption looks like that: $$Dec(c_i) = c_i.second - c_i.first\cdot s$$

Intuitively, i think, that this scheme is vulnerable, but i can't see why. I will be glad to have a hint to understanding this problem.

Many thanks in advance to your great community!
(sorry for any grammar mistakes, English is not my native language)