The organization has invited potential applicants to solve a visual code posted at website.
The challenge has also been "seeded" to social media sites, blogs and forums.
A spokesman said the campaign aimed to raise the profile of GCHQ to an audience that would otherwise be difficult to reach.
"The target audience for this particular campaign is one that may not typically be attracted to traditional advertising methods and may be unaware that GCHQ is recruiting for these kinds of roles," the spokesman said.
"Their skills may be ideally suited to our work and yet they may not understand how they could apply them to a working environment, particularly one where they have the opportunity to contribute so much."
The competition began in secret on November 3rd and will continue until December 12th.