Samsung has released the Android 4.1.2 Jelly Bean update for Galaxy S3 Mini with model number GT-I8190. The new official firmware is currently available as an Over-The-Air (OTA) update as well as through Samsung KIES.
Samsung Galaxy S3 Mini I8190 users who have already updated their devices with Android 4.1.2 Jelly Bean, may root their devices using the tutorial below.
Disclaimer: Rooting voids the warranty of your device. We and the developer of this rooting procedures shall not be held responsible if any undesirable outcomes may happen to your device. use of this root guide is solely at your own risk!
Pre-Requisites
1. Install USB Drivers for Samsung Galaxy S3 Mini on your computer.
2. Enable USB Debugging Mode on the device by navigating to Settings >> Developer Options.
3. Make a backup of all your important data before following the guide.
4. The Galaxy S3 Mini should be factory unlocked and not locked to any particular carrier.
5. Ensure the battery of the device is charged more than 80 percent.
6. The ClockworkMod Recovery (CWM) and TWRP Recovery files works only on Samsung Galaxy S3 Mini I8190. Applying these on any incorrect variant might brick the device. Verify the model number of the Galaxy S3 Mini by selecting Settings >> About Phone.
Files Required
1. CWM (Mirror Link), CWM Touch (Mirror Link) or TWRP (Mirror Link)
2. SuperSU (UPDATE-SuperSU-v1.75.zip) or the latest version
3. Odin 3.07 (Mirror Link)
Steps to Install CWM/TWRP Recovery on Samsung Galaxy S3 Mini I8190
Step 1: Extract the Odin 3.07 .zip file using WinRAR/7-Zip/WinZip or any other extracting tool.
Step 2: Copy the SuperSU .zip file to the external SD card of your phone.
Step 3: Switch off the Galaxy S3 Mini. Then boot the device into Download Mode by pressing and holding Volume Down, Home and Power buttons together until a construction Android robot icon with a warning triangle appears on screen. Now press the Volume Up button to enter Download Mode.
Step 4: Run Odin on the computer as an Administrator.
Step 5: Connect the Galaxy S3 Mini to the computer using USB cable while it is in the Download Mode. Wait until Odin detects the device. When the device is connected successfully, the ID: COM box will turn to light blue with the COM port number. Additionally, the successful connected will be indicated by a message saying Added.
Note: If the Added message does not appear, then try re-install USB drivers or changing the port on the computer.

Step 6: In Odin, click the PDA button and select the CWM/TWRP Recovery .tar.md5 file. Verify that Auto Reboot and F. Reset Time checkboxes are selected. Also, ensure the Re-Partition option is NOT selected.
Step 7: Double-check and click Start button in Odin. The installation process will now begin.
Step 8: Once the installation process is completed, your phone will restart and soon you will see a PASS message with green background in the left-most box at the very top of the Odin. You can now unplug the USB cable to disconnect it from the computer.
Steps to Root Galaxy S3 Mini I8190 by Installing SuperSU via CWM Recovery
Step 9: Reboot the device into CWM Recovery by pressing and holding Volume Up, Home and Power buttons together.
Step 10: In CWM Recovery, select "install zip from sdcard" and then select "choose zip from external sdcard". Now, navigate to the SuperSU .zip file that you copied earlier and confirm the installation by selecting Yes.
Step 11: Once installation process is completed, return to main recovery menu and select "reboot system now".
Samsung Galaxy S3 Mini I8190 running on Android 4.1.2 Jelly Bean is now rooted successfully. You can now install any app that requires root permission. Verify the root status of the device by downloading Root Checker app from Google Play Store.
Thank's to all fellas at XDA who first tried out this rooting technique. This root method was originally posted via XDA-Developers original thread.
Click here to know How to Update Galaxy S3 Mini I8190 with official Android 4.1.2 Jelly Bean firmware.

Samsung has just released the latest Android 4.1.2 Jelly Bean update with firmware build number I8190DXAML1 for Galaxy S3 Mini users in Southeast Asian countries. The new official firmware is currently available as an Over-The-Air (OTA) update as well as through Samsung KIES.
Firmware details
PDA: I8190DXAML1
CSC: I8190QLBAML1
MODEM: I8190DXAML1
Version: Android 4.1.2
Build date: Dec 27, 2013
Regions: Indonesia, Malaysia, Philippines, Singapore, Vietnam, Thailand
Samsung Galaxy S3 Mini I8190 users who missed the OTA update notification or unable to access the update over Samsung KIES, may follow our step-by-step guide to install the update manually.
Pre-requisites
1. Install USB Drivers for Samsung Galaxy S3 Mini on the computer.
2. Enable USB Debugging Mode on the phone by navigating to Settings >> Developer Options.
3. Make a backup of all your important data before following the guide.
4. The Galaxy S3 Mini should be factory unlocked and not locked to any particular carrier.
5. Ensure the battery of the phone is charged more than 80 percent.
6. The Android 4.1.2 DXAML1 Jelly Bean firmware is compatible only with Samsung Galaxy S3 Mini (GT-I8190). Applying this to any incorrect variant might brick the device. Verify the model number of the Galaxy S3 Mini by navigating to Settings >> About Phone.
Files Required
1. Android 4.1.2 DXAML1 Jelly Bean Firmware (Mirror Link)
2. Odin 3.07
Steps to Update Galaxy S3 Mini I8190 with Android 4.1.2 DXAML1 Jelly Bean Firmware
Step 1: Extract Firmware zip file using WinRAR/7-Zip/WinZip or any other extracting tool, you will get a .tar.md5 file and some other files (optional).
Step 2: Extract Odin zip file and you will get Odin3 v3.07.exe along with few other files.
Step 3: Switch off the Galaxy S3 Mini.
Step 4: Boot the device into Download Mode by pressing and holding Volume Down, Home and Power buttons together until a construction Android robot icon with a warning triangle appears on screen. Now press the Volume Up button to enter Download Mode.
Step 5: Run Odin on the computer as an Administrator.
Step 6: Connect the Galaxy S3 Mini to the computer using USB cable while it is in the Download Mode. Wait until Odin detects the device. When the device is connected successfully, the ID: COM box will turn to light blue with the COM port number. Additionally, the successful connected will be indicated by a message saying Added.
Note: If the Added message does not appear, then try re-install USB drivers or changing the port on the computer.

Step 7: Select the following files in Odin from extracted I8190DXAML1 firmware folder:

• Click the PDA button and select the .tar.md5 file.
• Click the Phone button and select file with MODEM in its name. Ignore if there is no such file.
• Click the CSC button and select file with CSC in its name. Ignore if there is no such file.
• Click the PIT button and select the .pit file. Ignore if there is no such file.

Step 8: In Odin, enable Auto Reboot and F. Reset Time checkboxes. Also, ensure the Re-Partition checkbox is chosen only if a .pit file was chosen in the previous step.
Step 9: Verify everything suggested in Step 7 and Step 8 is done correctly. Then hit the Start button to begin the installation process.
Step 10: Once the installation process completes, your phone will restart and soon you will see a PASS message with green background in the left-most box at the very top of the Odin. You can now unplug the USB cable to disconnect your device from the computer.
Samsung Galaxy S3 Mini I8190 should be now updated with official Android 4.1.2 DXAML1 Jelly Bean. Head over to Settings >> About Phone to verify the software version running on the device.
Note: If Galaxy S3 Mini I8190 stuck at booting animation after Step 10 then go to recovery mode (press and hold the Volume Up, Home and Power buttons together) and perform Wipe data/Factory Reset and Wipe Cache Partition task before selecting reboot system now function. In recovery mode use Volume keys to navigate between options and Power key to select the option. Please note, performing wiping task will erase your device internal memory data and not external SD card data.

Click here to know How to Root Galaxy S3 Mini I8190 running on Android 4.1.2 Jelly Bean firmware.