FWIW, current version (on CPAN) of IO::Socket::SSL is 2.015. Net::SSLeay vers. is 1.68. And LWP::Protocol::https is 6.06.
Can you update your modules? Seems to me we've recently (March) heard about an exploitable weakness (aka "FREAK" and/or "the heartbleed bug") in some versions of SSL.
But this may be a red herring. Your problem statement, "Since two days i can't generate bitly url" to a native English speaker strongly suggests that prior to two days ago, you were able to "generate bitly url."
If that's actually the case here, what changed (other than the inability to build short URLs)?