Apple - Reliable way to downgrade iPad 2 to iOS 6 without SHSH

Apple is currently signing iOS 6.1.3 for the iPad 2 and iPhone 4S. Simply download the appropriate IPSW and Shift+Restore (Option+Restore on MacOS) in iTunes. The signing status of iOS 6.1.3 for these two devices is unlikely to change because they are installed with iOS 5 out of the box. iOS 5 on these devices cannot update over-the-air to any iOS above 6.1.3 due to some technical limitation. So the update path for an out-of-the-box device over-the-air is iOS 5 -> iOS 6.1.3 -> latest iOS.

Note: At the time I answered, only the OTA for 6.1.3 was signed (because the limitation doesn't exist when updating through IPSWs). For some unknown reason, Apple now signs the IPSW for 6.1.3 (perhaps to be consistent). I don't know when this changed and I cannot find any information about it. The whole reason OdysseusOTA exists is because only the OTA used to be signed. Below you can find my original answer according to the circumstances of early 2016.

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The lowest version you can go with an iPad 2 and without SHSH blobs is iOS 6.1.3. This is because Apple still signs OTA (over-the-air) iOS firmware for this version. You can use OdysseusOTA to downgrade to this version, but only if you're jailbroken. If you're not, then you can't. And only iOS <= 9.0.2 can be jailbroken currently. Read my answer on this question for more information about OdysseusOTA.

All tools that claim to be able to downgrade are simply forks of Odysseus and/or its process. Odysseus is the only tool that can do such things at this time.

For a video explaining how to use OdysseusOTA, watch this YouTube video from its author.