A PDF-file not being specific to Windows is unlikely to yield its password protection status in a Windows extended file attribute and if it did, at the very least the OS must have opened it to peek inside!.

CountZero

A program should be light and agile, its subroutines connected like a string of pearls. The spirit and intent of the program should be retained throughout. There should be neither too little or too much, neither needless loops nor useless variables, neither lack of structure nor overwhelming rigidity." - The Tao of Programming, 4.1 - Geoffrey James

Beware that this approach has serious caveats: if your data can be real/pseudo random or compressed data, then it will also look like than a pile of random bits, so for example you won't be able to distinguish between a simple and an encrypted rar/zip file. (And don't forget that simple looking document formats can use compression internally.)

For the concrete implementation search for the chi square test on CPAN (I haven't looked but I'm almost sure you'll find some implementation) and try to experiment with it whether it can be good enough for your purpose.

++ for thinking outside the box

+1 because it's a really neat idea, but your caveat applies more widely than you might think; most PDF content is compressed, for example, and MS Office documents in the 2007+ format are literally ZIP files. So in practice it might be more useful to go straight for per-format tests.