In previous versions of Arduino, the limiting 8-bit microcontroller board, it seems that implementing HTTPS (not merely HTTP) was almost impossible. But the newer version of
IMHO Arduino (including the DUE) is the wrong tool for heavy and/or encrypted web based communication. I would strongly suggest to look for more appropriate hardware in the same size and price range. As soon you get into https you are close enough to also want a lot of the other stuff that real operating systems provide. With other words I suggest to go for something like the Raspi. Similar size and prize but way more powerful, especially it can run Linux. --> HTTPS becomes simple.