Non trivial bijective maps from $\mathbb{N}^{3}$ to $\mathbb{N}^3$

Question

The main reason I ask this question is because I want to implement an information scrambling or permutation algorithm on a $3$-D cube. The input to my bijective map would be the coordinates $(x,y,z)$ of the cube and the action of the map would be to output a triplet $(x',y', z')$ belonging to the cube. What i want further is that the mapping should seem random but should be deterministic. I guess chaotic maps such as Arnold Cat map does this job. But are there any other maps where i can look for or can we construct such bijective mappings. Any suggestions would be helpful. can simebody help me out with this?