Typo-Protected Public Keys

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When users type in public keys (such as the 53-characters of a GNS zone), they might make typos. The usual way to fix typos is to add a checksum, further increasing the length of the sequence that has to be typed in.
We can fix this by including the checksum of the public key in the public key itself, simply by trying new private keys until the corresponding public key happens to have a checksum (over the other bits) in the bits designated for the checksum. If a checksum is 16 bits, we would only need to try 216 keys. The basic idea of brute-forcing keys to match a particular pattern was proposed before for creating "vanity" public keys, but this might be another practical variant.

Acknowledgements
The idea popped up in a discussion on the need for short public keys for GNS with Dan Bernstein and Tanja Lange at 30c3.