Is the One Time Pad secure in additive Rings? Let's assume all operations are done on $\mathbb{Z}_p$ where $p$ is a large non-prime number. 

To mask a value $a$, we do the following:


Pick a uniformly random value: $r$, from the ring.
Do as follows: $c= r+a \bmod p$.


Question: Is the above one-time pad secure?
The classical xor-based one-time pad can be generalized to finite groups.

Let $(G,*)$ be such group with order $p$ and $*$ is the group operation(like the xor). 
The message, the pad and the ciphertext are elements of $G$.

Now to encrypt a message $m \in G$, choose $k \in G$ uniformly at random and set $c = m * k$. One of the security proofs of the one-time pad consists of showing that $c$ does not give any information on $m$(i.e to find $m$ we might as well pick a $c'$ at random and ignore $c$ completely).

More precisely, if $M$ is a random variable for messages distributed somehow, $K$ is a uniform random variable for the keys, and $C = M*K$ the random variables for the ciphertexts. 

What we need to show is that $C = M*K$ is independent of $M$. i.e $C$ does not give any information on $M$.

Proof: We want to show that $P_{C|M}[c|m] = P_C[c]$. First, It's easy to see that $C$ defined as above is uniform(i.e $P_C[c] = \frac{1}{|G|}$). Next observe that $P_{C|M}[c|m] = P_{K|M}[c*(m)^{-1}| m] = P_K[c*(m)^{-1}] = \frac{1}{|G|}$. This follows form the fact that $K$ and $M$ are independent. 

Therefore we showed the 'perfect secrecy' property of this constriction.