Odysseus

Odysseus is a tool by xerub to downgrade/upgrade/restore specific 32-bit devices to specific firmwares. This method only works if you are jailbroken, have valid SHSH Blobs and a valid APTicket.

How it works
It uses a tool included in winocm's ios-kexec-utils called kloader. kloader is a CLI that you can run on an iOS device which loads an (decrypted) image. It allows you to bootstrap a pwned iBSS and then initiate a restore using idevicerestore with a custom IPSW, which allows you to downgrade/upgrade/restore to the firmware of your choice. There are firmware bundles that allow you to restore each device to each iOS. If the bundle for the device and iOS you want to get does not exist, the downgrade/upgrade cannot happen. Each bundle is made by grabbing the Keys/IVs from the original firmware. They are posted here.

Links

 * OdysseusOTA, a derivative tool for downgrading certain devices to iOS 6.1.3
 * OdysseusOTA2, like OdysseusOTA, but for downgrading to iOS 8.4.1
 * Official page
 * GitHub
 * Unofficial bundles
 * Unofficial bundles