How to downgrade samsung galaxy s3 4.3 to 4.1.1
You need to download Odin for Windows, then download 4.1.1 firmware for your model. If it downloads in .zip files you need to decompress. After decompressing the file you will see a .tr.md5 file, this the firmware itself, this is the one you are going to flash. Also you need to have the drivers for you model device already install, you don't have them yet, install Kies and after installation reboot your computer.
Now, turn off the phone, and to make sure nothing goes wrong let's hard reset the phone first by pressing and holding at the same time the volume up, home and power key, until you see the recovery page. Now choose Wipe data/ factory, press the power key to select, scroll down to yes and press the power key.
Do the same for Wipe Cahe. Once that is done, go back to the recovery main page and select Reboot. Turn off the phone again, this time press Volume down, home and power key at the same time until you see the Android logo, this is the download page. Press the volume up rocker to start the download, open Odin, connect the USB cable to the GS3. Notice at top of Odin different squares, at the bottom of the first square it should display the com you're using, if it doesn't show anything then you need to re-install the drivers for your device.
Once it shows the com number you are ready to go, make sure that Auto Reboot and F. Reset Time are checked, leave the others uncheck. Now click PDA, or in latex versions of Odin is AP, search for the .tar.md5 files that was decompress and choose it. Now click start at the bottom of Odin and wait until it shows Pass. Now you should be in 4.1.1 stock ROM. If you are rooted, this will unroot your phone.