Your public/private key pair are pretty straightforward public key
cryptography. If you can brute force one given the other, it wouldn't be
safe to use it over the wire in the first place.
The point of having a passphrase is that some really paranoid people want to
have both a key AND a password to get into their box. A password in and of
itself isn't all that secure, since any human memorizable password contains
so little information it would be trivial to break computationally via brute
force (if you can arrange a brute force attack, which login tries to prevent
with the ~3 second delay between attempts, but they still tend to get written
down, or people watch keystrokes over your shoulder...). And a key you carry
around with you on a floppy or propogate to multiple boxes can be stolen
(copied) from any of those places without you necessarily knowing about it.
In terms of brute forcing the key, the passphrase adds a fairly trivial
number of bits to the key. It's not "far" easer, an 8 character mixed case
nonsense password with numbers and punctuation mixed in is still less than 6
bits per character, or at best an extra 48 bits. You can select a 1024 bit
key if you want to be really really paranoid.
Not that it's worth it. Keys get exponentially more difficult to brute force
as the key length increases. I read part of a book a long time ago (might
have been called "applied cryptography") that figured out that if you could
build a perfectly efficient computer that could do 1 bit's worth of
calculation with the the amount of energy in the minimal electron state
transition in a hydrogen atom, and you built a dyson sphere around the sun to
capture its entire energy output for the however many billion years its
expected to last, you wouldn't even brute-force exhaust a relatively small
keyspace (128 bits? 256 bits? Something like that).
Somebody else here is likely to recognize the above anecdote and give a more
accurate reference. Book title and page number would be good...
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