Can you play upgrades on opponent's creatures? If so who 'controls' the upgrade?

E.g. (If legal) I play an upgrade that says "Your opponent's keys cost 2 more to forge." Does that make MY keys more expensive (b/c the upgrade is on his creature) or my opponent's keys more expensive (because I still control the upgrade even if it on is on their creature).

UPGRADES (p6):
Upgrades enter play attached to (i.e., partially overlapped by) a creature chosen by the player who controls the upgrade. Each upgrade remains in play from turn to turn and modifies the card to which it is attached.
If the card to which an upgrade is attached leaves play, the upgrade is discarded.
If an upgrade cannot attach to a card in play, the upgrade cannot enter play.

UPGRADES (p6):
An upgrade modifies the creature it is attached to and is not used independently of that creature.

I think the main part here is, yes, you can upgrade an opponents creature, and it is used as part of that creature. From your opponents creatures POV you are the opponent so that would make your keys cost +2. Of course if the card gives you 2 amber and you then kill the creature you attached it to it may still be an advantage to you.