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According to this heartbleed mass test, Stack Overflow is vulnerable whereas google is not.

I use my Google ID to logon to Stack Overflow (and some other sites). Do I need to change my password? In other words, is it the account that determines safety, or the service used?

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Yes as your password which is supposed to be encrypted with SSL tsl is readable in plain text and if you use the same password for both web applications (you shouldn't be anyway) an attacker has access to both.. However if you use googles openID to log in it goes through googles servers and you would be secure

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