I believe there is a misunderstanding here: If you want to do the oasis loop only, there are no special permits required. You can go from Cairo to Bahariya, on through the White Desert to Farafra, Dakhla and Kharga before reaching Luxor.
The tarmaced roads in the deep South(-West) of Egypt are often either military or need a permit (which is usually refused). Taking the road from Siwa to Bahariya requires a permit which you easily obtain if you either possess a sat-phone or follow other vehicles (mostly local tour operators) who do carry a sat-phone with them.
What is meant here, though, are expeditions off road to areas south of the 23 latitude.