Conditional entropy under a deterministic change of variables
= Conditional entropy under a deterministic change of variables
If $f(x,z)$ is a <bijection> of $x$ for every fixed $z$, then
$$
H(f(X,Z)\mid Z)=H(X\mid Z).
$$
In particular, translating a random variable in a finite group by a value determined by the conditioning variable does not change its conditional entropy.