Why haven't asymmetrical keys replaced passwords yet?

imagined security scheme:
1.Generate public/private key pair
2. share public key when creating account with whomever.
3. when logging in, account will send you random string
4. your browser will encrypt string with private key
5. account will decrypt string with your public key, if it is the same string they sent, you are authenticated.


@zacharius Minilock! Minilock! Minilock!

(I hope Minilock is actually as secure as it claims; I have no way of telling. I just love the idea of a super-short public key being my identity, and its matching super-short private key being automatically generated from my username and passphrase).


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