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Or would it just say we need to have your private key?

That's a great question because I don't particularly protect my private key.

In addition to the publicly available information, Bob knows his own private key.

I see you got a private key to this dump.

But it doesn't have the private key for any of the clients.

All parts of the private key must be kept secret in this form.

This is her private key and she will store it away in a secure place for later use.

Is that part of the private key in some way?

Well, they would never have the private key used to sign the certificate.

She can use her own private key to do so.