Can malware run on mobile devices? I don't see how it would be able to (given that those devices have fine-grained permissions and sandboxing). Unlike on a desktop OS, malware cannot access registry components; it also can't have root access unless the device is rooted. So does this mean current mobile OSes are immune to standalone malware?
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Yes. Just about every security feature of mobile operating systems is implemented in software. Complex software like that
can will have bugs. Some of these bugs will be exploitable to give unauthorized access. This applies to any sufficiently complex software, because programmers are no more superhumanly perfect than anyone else.