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Is there a benefit in measuring randomness? How would it be done?

Is there any benefit in determining how random a given file, stream, signal is? I guess this would be useful to determine If something is (poorly) encrypted To verify the proper encryption of a ...
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Is it possible to tell when a remote bitmessage client is creating new messages?

One StackExchange answer mentioned bitmessages can be traced, another said it is fully anonymous. The second link seems to cite Wikipedia but a user points out that this anonymous concept doesn't ...