I mean a password with enough entropy that also hasn't been used before and is not shared with anyone.

Do you know how they were owned? Is there a write up anywhere as this is a super interesting topic.

> but I do know people with strong SSH passwords that got owned up

By having their password brute forced, or some other means?

So for example, a 16 character random password is still vulnerable?

This is what I would like to know. If strong ssh passwords can be broken then there is a bigger issue than just turn off ssh password log in.

I am pretty sure that 16 character random passwords are safe from dictionary attacks, and I suspect you and I are using the same process for protecting these passwords. tptacek is leary of our process while you and I are leary of the certs getting out of our control without our knowledge. Security is a handful of trade offs.

For passwords "cryptographically strong" normally means a password with appropriate complexity and sometimes unpredictability requirements, since entropy is almost nonexistent in passwords.

Something out of pwgen or diceware would do though. If your application has length requirements that preclude the latter, make sure they are liable if you loose something due to preventing you from using a secure password. (No one can remember a short secure Password for more than a small handful of applications.)