How are the cipher, the key and the initial message (that is not encrypted) are releted?

Question

Suppose that $m$ is a message that someone player $i$ wants to send to a network of other players $j\neq -i$. The player to prevent his message from cheating by others uses an encyrpstion scheme. Say $$g:M\times Y \to X$$ denotes a cipher where $Y$ is the key and $X$ a code that makes the message to look random. The standard assumptions to be made are that $|Y|\geq |M|$ and $g(\cdot,y)$ is a bijection namely every pair of $(m,y)$ is associated with only one $x$. My question is how are the key $y$, the code $x$, and the message $m$ are associated? for example if we could make some operations among $g$, $y$ and $m$, what would that be? could we claim that $x\oplus y \underbrace{=}_{?}m$? or somehting like this?

Answer 1

Taking into account the book. I write here an example. Suppose, that we have a mechanism of communication $\mathcal{M}=(g,h)$ such that $\mathcal{M}$ is defined over $(Y,M,X)$, where $Y$ is the key, $M$ the message and $X$ the cipher spaces respectively. To simplify the problem even more I assume that $Y=M=L=\{0,1\}^l=G$ instead of an arbitrarily finite field $\mathbb{F}^n$ and write below

$$g(y,m)=x,\quad\text{is the encrypted message, which by definition equals $x$}$$

$$h(y,x)=m,\quad\text{is the decrypted message, which by definition equals $m$}$$

So, indeed $(y,x)$ is defined to be associated with only one $m$ and hence $g(y,\cdot)$ is bijective by definition. To anser the question how are they associated, when someone knows both $x$ and $y$, then indeed $x\oplus_{G} y=m$

In order to decrypt the message we have that

$$h(y,x)=h(y,g(y,m))=y\oplus_G x=m$$

where $\oplus_{G}$ is the operation of $+$ as it is defined in the finite field $G$. And hence we have show that the calculation that you ask for, it holds by definition.

Answer 2

You can't really say much beyond what is true by definition, i.e. that $g(m, y) = x$. The problem as stated intentionally leaves $g$ minimally restricted, so there are any number of concrete functions that would satisfy the statement. Further, $M$, $X$, and $Y$ are also minimally restricted by being defined as sets, and as such you can't assume the existence of any such function $\oplus$.

Remember that the purpose of using such general definitions is to prove theorems with the broadest implications. Then you know that the results apply to any particular function that satisfies the conditions.