
Samsung Galaxy S III
The latest firmware for Samsung Galaxy S III (SCH-I535PP) is I535PPVRUBNF5 with Android 4.4.2 (KitKat), released on August 29, 2014 for 1 region (CSC). Every official firmware build for SCH-I535PP, newest first: one row per build with all regions (CSC) that received it.
Each firmware build has its own page at https://samfirm.shop/build/{PDA}, where {PDA} is the build number from the list above. Example: https://samfirm.shop/build/I535PPVRUBNF5

SCH-R970X

SCH-L710

SCH-I939D

SCH-I435

SCH-I545
You can find the firmware on the model: SCH-I535PP page.
Yes, all firmware files for SCH-I535PP are 100% official, original, and obtained directly from Samsung servers.
To flash SCH-I535PP, you need to download the appropriate firmware, put your device into Download Mode, and use the Odin tool to load the BL, AP, CP, and CSC files.
Downgrading is only possible if the bootloader version (binary bit) of the old firmware matches the current one. Always check the binary version before flashing SCH-I535PP.