The electronic surveillance organisation, the UK Government Communications Headquarters, is asking potential applicants to solve a code posted on a website.
It will direct potential candidates to the competition, hosted on an anonymous website, via sites such as Facebook and Twitter.
If the layers of code it has set are cracked, applicants will be presented with a keyword to enter into a form field.
They will then be re-directed to the GCHQ website, where hopeful hackers will find details of the types of roles which could reflect their skills.
The aim is to attract candidates who might not apply through more conventional channels.
A spokesman for GCHQ said: "Traditionally, cyber specialists enter the organisation as graduates.
"However, with the nature of computer threats ever-changing, it is essential we allow candidates who may be self-taught, but have a keen interest in code-breaking and ethical hacking, to enter the recruitment route too.
"Our target audience is not typically attracted to traditional advertising methods and may be unaware that GCHQ is recruiting for these kind of roles."
The organisation says any potential codebreaking recruits will be rigorously vetted and any applicant who has previously hacked illegally will be rooted out.
GCHQ said the sort of person who takes on the online challenge is likely to have the sort of skills it looks for in applicants.
"Their skills may be ideally suited to our work," said the organisation. "And yet they may not understand how they could apply them to our working environment, particularly one where they have the opportunity to contribute so much."
GCHQ has used unconventional recruitment techniques in the past.
In 2009, the organisation launched a six-week campaign featuring advertisements on the Xbox LIVE network, which appeared during video games including "Call of Duty" and "Assassin's Creed".
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