I found myself wondering today, how much security is lost if you take a plaintext - assume that its content, including any metadata is unknown to an attacker, for example it may be random data - and encrypt it with multiple keys (not chained) and give all resulting ciphertexts to an attacker, how much higher is the probability of an attacker discovering the plaintext, vs only having a single ciphertext. 

An example:

Take plaintext $P_1$ and encrypt it with $K_1$, and send the resulting ciphertext $C_1$ to attacker $A_1$.

In addition: take the same plaintext $P_1$ and, and encrypt it with $K_1$, giving $C_1$. Then take $P_1$ and encrypt it with $K_2$ giving $C_2$, then $K_n$ giving $C_n$ and send all $n$ ciphertexts to attacker $A_2$.

How much more likely is attacker $A_2$ to discover the value of $P_1$, than $A_1$?


Question: Why is it best to use a prime number as a mod in a hashing function?
Answer: If your hash function is of the form $h(k)=a\times k \mod m$ where $m$ is prime and $a$ is chosen at random, then the probability that 2 distinct keys hash to the same bucket is $1\over m$. So for $m=1009$, $Pr\{h(x)=h(y), x\neq y\}=0.00099108027$ which is very small.

This scheme is known as: Universal Hashing.