For your convenience:
The researchers pointed out that the vulnerability cannot be exploited remotely. An attacker can trigger the issue by providing crafted inputs to applications that employ these [syslog] logging functions [in apps that allow the user to feed crafted data to those functions].
This is a privilege escalation.
This may be difficult to exploit in practice - I don’t think most user applications use syslog.
You still need some privileged process to exploit. Glibc code doesn’t get any higher privileges than the rest of the process. From kernel’s point of view, it’s just a part of the program like any other code.
So if triggering the bug in your own process was enough for privilege escalation, it would also be a critical security vulnerability in the kernel - it can’t allow you to execute a magic sequence of instructions in your process and become a root, that completely destroys any semblance of process / user isolation.
