I live in France and I would like to know if it was legal to attack the Virtual Private Server I am renting in the US.
- Consider that the provider doesn't know it's you that is attacking your own VPS, so they could start an investigation, which is expensive, and might also incur fines if they alert the police forces
- Some attacks (DDOS for instance) will necessarily affect also other users, not only you.
Bottom line, I'm definitely not a lawyer, but I'm pretty sure that if you don't have their permission, it's illegal everywhere.
Ask your provider, and if he refuses to grant you permission, find a provider which specifically allows such a thing (I guess they must exist somewhere, after all, white hats must do some tests sometimes…).
Keep in mind that I think it's unlikely you'll find a VPS which allows such a thing, since you would risk crashing the whole machine, it's much more likely you'll end up needing a dedicated server.