Why is entropy of three variables equal to entropy of two variables?

Question

For each X - messages Y - encoded messages Z - keys, next statement is true:

$ H(X,Y,Z) = H(X,Z) = H(Y,Z) $

Why is that true ?

Answer 1

The encryption mapping $E$ must be well defined for each fixed $Z;$ thus $E:(Z,\cdot):X\mapsto Y$ is such that for each fixed $Z,$ $X$ uniquely determines $Y$ which is exactly the condition $H(X,Y,Z)=H(X,Z).$

The encryption mapping must be invertible for all possible key values $Z$ otherwise the cipher is useless in recovering the message $X$ from the ciphertext $Y$. Now apply the same argument to $E^{-1}(Z,\cdot):Y\mapsto X$ for fixed $Z$ to get $H(X,Y,Z)=H(Y,Z).$

Answer 2

We have, \begin{eqnarray} H(X,Y,Z) = H(X,Z) + H(Y|Z,X) \\ = H(Y,Z) + H(X|Z,Y) \end{eqnarray}

As $Y$ is a direct function of $(X,Z)$, $H(Y|X,Z) = 0$. As the earlier answer has pointed out, Given the key $Z$ and encrypted message $Y$, we should be able to determine $X$, otherwise, it is a useless encryption algorithm. Hence we also have $H(X|Z,Y) = 0$, which completes the arguments.