Allow to upload a PGP public key, and reset request page would just return an encrypted PGP message containing a link to reset the password.

Assumes: - people are less likely to lose their PGP key, than random password to a random website. - people have PGP keys - PGP key doesn't contain email address (it does).

Anyway, it would be reliable, and it doesn't need giving third party online service access to all your online accounts.