# 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?

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In the future, please tag homework questions as such. –  pg1989 Nov 29 '13 at 22:42
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 $Z$.

As a help towards answering the question whether scheme $M + K$ mod $m$ is perfectly secure, when $m$ is 26 this scheme is also called the Caesar Cipher.

The definition of perfect security is:

Claude Shannon proved, using information theory considerations, that the one-time pad has a property he termed perfect secrecy; that is, the ciphertext C gives absolutely no additional information about the plaintext. This is because, given a truly random key which is used only once, a ciphertext can be translated into any plaintext of the same length, and all are equally likely.