You're so very close.
You're looking for the Rings of the Master books, by Jack Chalker.
They're set a few hundred years in the future; in the backstory, mankind developed a huge computer system, the Master System, and gave it the prerogative to protect humanity. In true SF style, it decided to do this by sending
most of Earth's population to form colonies on distant planets, and splitting the remaining population on Earth into various cultures, all set firmly in a pre-industrial period (and kept that way even more firmly)
Several of the heroes are from the North American cultures, which have mainly been formed into pre-Columbus
Native American tribes; and the Master System has a secret back door built in, which requires the use of five (not twelve) rings in a specific pattern in a particular place to activate it.