[AT&T, Rogers, Bell, Telus Samsung Galaxy S III] spasch:
I did mine a long time ago. After reading quite a bit you may be right for the time being. The best I have found without a risk of a brick is this:

There is a lot of confusion about unlocking in this thread and elsewhere. Your S3 can still be unlocked after the 4.3 update, just keep in mind that most threads either have inaccurate information or miss essential steps.

Unlock codes and procedures often change or play musical chairs in major software updates. Prior to the problematic 4.3 update you could unlock your S3 with the cited procedure in this thread. This is a permanent unlock and would survive firmware changes. This only applies to touchwiz ROMs e.g. North American factory ROMs or custom ROMs that preserve the dialer codes. This unlocking procedure is also distinct from the carrier specific unlock codes or commercial unlockers who either have access to the official database or have figured out the key algorithm.

If your phone has not taken the 4.3 update yet:

Install the 4.1.1 stock touchwiz ROM and use this procedure to unlock. You can then revert back to your favourite firmware version after your phone has been unlocked.

Dial the following code on the phone keypad: *#197328640#

Main Menu > [1] UMTS > [1] Debug Screen > [8] Phone Control > [6] Network Lock > Options [3]Perso SHA256 OFF

After choosing this option, wait 30 seconds, then go back one step by pressing the Menu button then select back

Now you are in [6] Network Lock then choose [4] NW Lock NV Data INITIALIZE

You should see a message that says

SHA256_ENABLED_FLAG[0]

SHA256_OFF => Not Change

Choose [1] SHA256_ENABLED_FLAG

Wait 30 seconds, then press Menu > Back. Do not use the back key, choose back from the menu popup.

You should now see a message that says

MENU NOT EXIST

PRESS BACK KEY

Current Command is 118640 (if you don't see this message.. skip to alternate step below)

Note that the exact Command number may vary.. don't worry about that.

Wait another 30 seconds, then press Menu > Back. Do not use the back key, choose back from the menu popup

You should see a message that PERSO_NV_INIT is processing, then PERSO_NV_INIT is SUCCESSFUL

Now return to the SHA256 menu and turn Perso SHA256 back on

Back our of all menus and enjoy your unlocked S3

Alternate step. If you didn't receive the MENU NOT EXIST message, then use the back option in the menu popup until you see a menu what includes [6] Common.

Click on [6] Common and choose [6] NV REBUILD

You should now see a message that says:

Golden-Backup exist

You can Restore Cal/NV

Choose [4] Restore Backup

Your phone may freeze or change screens briefly, then should reboot

If you run into errors, look at this link for advice on how to proceed: How to unlock.. (techbeasts.com)

Enjoy your unlocked S3

If you have the 4.3 update:

Now you know that updates are evil or at least mess up your plans.

The unlock procedures have changed somewhat now and it seems that there are now several alternate procedures depending on your carrier.

Gambit #1

If you tried the above and got half way to find that menu item [4] NW Lock NV Data INITIALIZE was missing, then try the following. Note that menu item [4] may also be missing if your phone is already unlocked. So you might want to try a SIM from another carrier to confirm whether your phone really needs to be unlocked.

Dial the following code on the phone keypad: *#197328640#

Main Menu > [1] UMTS > [1] Debug Screen > [8] Phone Control > [6] Network Lock > Options [3]Perso SHA256 OFF

After choosing this option, wait 30 seconds, then go back one step by pressing the Menu button then select back

Now you are in [6] Network Lock and presumably you now find that there is no [4] NW Lock NV Data INITIALIZE menu item

If so, then press Menu > Back. Do not use the back key, choose back from the menu popup.

Now this is the step that most threads miss, tap on SHA256_ENABLED_FLAG.

This item may look like a message rather than a menu choice i.e. may not be enumerated. Tap it anyway

Now you may see one of several things.

If it appears that you are back to the starting menu, choose [1} UMTS again and follow the original steps and you should find that the missing menu item is now present, although it may be labeled NW Lock NV Data INITIALIZE or NV REBUILD. Or you may find yourself on a page that already has the initialize or rebuild option.

In either case, choose INTIALIZE or REBUILD, followed by the remaining steps as originally described.

If you get a message asking you to reboot your phone to finish, then do that instead of the remaining steps.

I'd recommend that after you are done that you return to the service menu again and ensure that SHA256 was re-enabled. The SHA256_ENABLED_FLAG should = 1 (not 0). If the flag status is 0, then choose the Perso SHA256 ON option.

Enjoy your unlocked S3

Gambit #2

If the above does is still missing key menu items on your phone, then try this alternate dialer code:

*#27663368378#

And then proceed as before.

However, if you encounter the following message:

Golden-Backup exist

Then follow the procedure in this link instead: How to unlock.. (techbeasts.com)

Enjoy your unlocked S3

Gambit #3

If you still have a stubborn phone, try the following dialer code

*#0011#

This will look like an informational screen with no menu choices.

Don't fear, the menu you want is just hidden. Continue with the steps in this link: How to unlock.. (unlockfaq.com)

Enjoy your unlocked S3

Oh no.. that didn't work either!

If all else fails, contact your provider and tell them that you will be traveling in elsewhere e.g. Europe and need to have your phone unlocked. Most carriers will provide you with a network (carrier) unlock code in this situation. Typically without charge, but it will vary by carrier. Or you can avail yourself of a commercial unlocking service. Google will find a googol [sic] of them for you.

Note that you need to insert a valid, non-blacklisted SIM from another carrier to be prompted to enter a network unlock code. It need not be active, just not blocked i.e. reported as lost /stolen. If you have any difficulties entering the code, have a look at this link.

Good luck

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