I need my Arduino to connect to my web server through HTTPS, can the Arduino handle it?
It only has to act as a client.
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It sounds like a better way to do it would be to have a PHP script act as handoff server (maybe protected by originating IP address) to access the HTTPS site and provide the data over a less processor-intensive protocol.
However, another answer suggests that we are working towards a solution to your issue, but it's not quite there yet (~12 seconds to encrypt a packet).