I know that there are a lot of asymmetric algorithms out there that use the public key to encrypt and the private key to decrypt. I was thinking about having exactly the opposite.
I do also know that the current way of doing it is sending it in plaintext and adding a signature to verify that its coming from the correct sender. But for me it sounds like it is just some sort of "overhead" with an additional signature to check and transmit.
In my mind it would be cool to say: If you can decrypt it, it is the right sender and if you can't, someone must have altered it / is not trustworthy.
If there isn't such thing, why is that exactly? Because the signature "method" is the way to go?