Bash is a Unix shell written by Brian Fox for the GNU Project as a free software replacement for the Bourne shell (sh). Most Unix-based operating systems today use bash as the default shell including Linux and Mac OS X, and additionally bash has been ported to many other systems including Windows (with software such as Cygwin) and Android. The fact that bash is so widely used means that this is an extremely significant vulnerability.
This new Shellshock vulnerability potentially allows attackers to execute arbitrary shell commands on your machine. But the even scarier thing about this vulnerability is that it's extremely old, meaning that it could have potentially been exploited for years now (and may have even been exploited without having been revealed). Since the vulnerability is so old, it's quite widespread, which means that it will likely still be seen unpatched in many systems for quite some time.
For more info about checking if you're vulnerable and patching your system, continue here: Are You Shellshocked?"
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