Earlier quoted context omitted.
You generally don't want to layer encryption like that. It apparently really does introduce new kinds of attacks, which has been observed in the real world. The pattern typically used for this is that the key for the high-speed symmetric encryption is split into multiple parts, each of which is encrypted with a separate public key system. One classical, one (or two, now?) with a post-quantum algorithm. As long as eac…
Running TLS over TLS is fine, or ssh over ssh, or ssh over TLS, or so on. Otherwise the bad guy would just put the TLS traffic they intercepted from you, through their own TLS tunnel and somehow acquire more information. In the early days of SSL there were cross-protocol information leaks if you used the same key or related keys for different protocols or protocol versions. In the DROWN attack, I can get some ciphert…
Here's Wikipedia: https://en.wikipedia.org/wiki/Multiple_encryption
I'm no expert here, but if I understand Wikipedia correctly:
* Be sure to use distinct keys and IVs for each individual layer.
* Be aware that encrypting ciphertext could lead to a known-plaintext attack on the outer cipher, if the inner ciphertext starts in a standard way (file header etc.)