How to Fix Game Center on iOS 6

1. Install signed bundle certificate from the following link: http://tlsroot.litten.ca
2. Install the application iFile from cydia.
3. Navigate to the directory: /var/mobile/Library/Preferences.
4. Look for any *folders* with names extremely similar to the following and delete or move them: purplebuddy, accountsettings, itunesstored, appstore, gamecenter, iTunesStore, and gamed.
5. Next, go to the directory: /var/mobile/Library/Caches and repeat the same process. However, instead of folders, you will be dealing with .plist *files*.
6. After deleting or moving these files, you need reboot (not respring) your device.
7. Then, go through the setup screen as usual.
8. When you reach the Apple ID login screen, sign in correctly, as you now have the required certificates installed.
9. For the final step, open Game Center and check if it loads. If it does not, please attempt the entire process again from the beginning.
10. Credits to calvink19 for originally writing this guide (I know this is not a "step" but it's worth mentioning here)