One thing I have a habit of doing since I work in the IT Security industry myself, is check not only a website's SSL certificate, but I also like to see how well they implemented it. The tool I often use to test this sort of thing is SSL Labs.
You see, not all implementations of SSL are secure. There are obsolete ciphers available. Plus, older implementations of SSL have been found to be vulnerable to various attacks. It's one of the reasons why PCI-DSS is forcing all companies that handle credit card transactions to disable everything except TLS 1.1 and TLS 1.2 in June this year.
I was curious on how well the cryptocurrency exchanges I use for various reasons stacked up. So I decided to test their websites using SSL Labs. Here are the results:
ECC is able to provide the same cryptographic strength as an RSA-based system with much smaller key sizes. For example, a 256 bit ECC key is equivalent to RSA 3072 bit keys (which are 50% longer than the 2048 bit keys commonly used today). The latest, most secure symmetric algorithms used by TLS (eg. AES) use at least 128 bit keys, so it makes sense that the asymmetric keys provide at least this level of security.