# Is this encryption scheme perfectly secure?

1. Let $m = 6$, and let $\mathbb{Z}_m$ denote the set $\{0,…,m-1\}$. Let $X \mod m$ denote the remainder obtained when dividing $X$ by $m$.

(a) Consider the symmetric encryption scheme in which the encryption of message $M \in \mathbb{Z}_m$ under key $K \in \mathbb{Z}_m$ is $M+K \mod m$. Is this encryption scheme perfectly secure? Why or Why not?

I'm having a lot of trouble understanding why the "X mod m" and all of the other modulo operations are relevant. I know it says let it denote the remainder obtained when dividing $X$ by $m$, but… the remainder of what? Why would the remainder be relevant?

• Please consider writing the text by hand, not just dumping a picture, because it makes search easier (and someone else is more likely to benefit from your question). Cheers – rath Nov 29 '13 at 22:48
• @rath [+1] Nice idea, so I quoted the text of that image. (Sometimes I wonder how I spend my friday nights…) – e-sushi Nov 29 '13 at 23:12

The modulo operator keeps the result of the addition of $M$ and $K$ within the set $Z$.
For example, if $m$ is 10, $M$ is 6 and $K$ is 5, $M + K$ would be 11 which is no longer in the set $Z$. Taking 11 mod 10 results in 1 which is in the set $Z$.
An interesting special case is perfect security: an encryption algorithm is perfectly secure if a ciphertext produced using it provides no information about the plaintext without knowledge of the key. If E is a perfectly secure encryption function, for any fixed message m there must exist for each ciphertext c at least one key k such that $c = E_{k}(m)$. It has been proved that any cipher with the perfect secrecy property must use keys with effectively the same requirements as one-time pad keys.