
Samsung Galaxy S 3 (Verizon)
The latest firmware for Samsung Galaxy S 3 (Verizon) (SCH-I535) is I535VRUDNE1 with Android 4.4.2 (KitKat), released on May 18, 2014 for 1 region (CSC). Every official firmware build for SCH-I535, 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/I535VRUDNE1

SCH-R970X

SCH-L710

SCH-I939D

SCH-I435

SCH-I545
You can find the firmware on the model: SCH-I535 page.
Yes, all firmware files for SCH-I535 are 100% official, original, and obtained directly from Samsung servers.
To flash SCH-I535, 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-I535.