Gnome 3 uses an authorization mechanism called polkit so the right way to run that executable as root is via pkexec:
pkexec allows an authorized user to execute PROGRAM as another user.
If username is not specified, then the program will be executed as the
administrative super user, root.
To authorize the user in the active session to run that command as root you'll have to define a new action (as per the above spec) via a
XML file under
/usr/share/polkit-1/actions , let's call it
<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE policyconfig PUBLIC
"-//freedesktop//DTD PolicyKit Policy Configuration 1.0//EN"
<message>Authentication is required to Change Keyboard Backlight</message>
Next step is to assign the commands to your keyboard shortcuts via GNOME keyboard settings:
pkexec /usr/local/bin/backlight inc
pkexec /usr/local/bin/backlight dec
(note: I used
/usr/local/bin/backlight as the path for my executable, you'll have to alter the action file and the two commands if your executable path is different)