I am working in ℂ3 in this question. (If it will make it easier, we can work in a bounded subspace.) Suppose you have, in each of the three complex planes whose Cartesian product make up the space in question, a set of points. (You do not have knowledge of generators of these sets, or whether such a generator exists, and you cannot assume connectedness in any of them.) I would like a relatively simple binary transformation, defined on any two sets of points, to combine them such that the method fulfills three conditions: (1) it is associative (for example, the Cartesian product or the operation of rotating the planes to be all on one plane would fulfill this) (2) it would be unique in the sense that if the sets A and B are combined to give C, then A cannot be combined with anything else to give C (For example, the Cartesian product would fulfill this condition, but the above rotation would not, since points could overlap) (3) it is a trapdoor transformation: that is, if A and B are combined to give C, then given only C it is very hard to get A and B back. (For example, the Cartesian product would not fulfill this condition, but the rotation method would.) Neither the Cartesian product nor the rotation method would fulfill all three conditions. Can anyone imagine such a transformation?