An application package from Geohot's webpage provides the root exploit
Being the owner of a less popular phone variant often means having to wait a little longer when it comes to support from the hacker community. For owners itching to root their Samsung Galaxy S4 Active devices, the wait is over. Infamous hacker Geohot -- most known for being the first person to carrier unlock the original iPhone -- has published a root method for the Galaxy S4 Active.
Some very vocal users on XDA Developers started a bounty, each individually pledging dollar amounts for the ability to root their beloved Galaxy S4 Actives. Not long after an official bounty forum thread was started, hacker Geohot answered the call by rooting his first Android device and making an application package available for download on his own website. Users simply visit the website, download the application directly onto a Samsung Galaxy S4 Active, and run "activeroot."
Credit to who actually found the root exploit has been questioned, however.
"Elite Recognized Developer" Jcase at XDA Developers has posted links to what he alleges is the same exploit, but found earlier this year. Jcase goes on to claim the exploit was previously ported to Android on more than one occasion, providing GitHub links as evidence. Since there is no obvious mention by Geohot on XDA Developers or his own website that he used an exploit found previously by someone else, some users have asked that Geohot at least give the other developer(s) credit, and at most share the monetary compensation provided by Galaxy S4 Active users.
Either way, without Geohot's work there would not be a root method available for the Galaxy S4 Active. To get Geohot's root application, visit the Activeroot website.
Source: XDA Developers Thanks, DJ Levy!