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.

- The biggest problem I see with this is not wanting to use unsecure computers because you would have to share your private key with that computer to log on, but the same could be said for passwords.

- convincing people to carry around a usb stick containing keys seems much easier than convincing them to choose and track dozens of hard to guess/bruteforcable passwords.

pretty sure this could all be done over http, with a browser extension to handle the authentication on client end


@zacharius I think that this in basically what WebAuthn is doing (with the addition of username-like features and per-website key pairs). developers.google.com/web/upda

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