There's been another interesting development since then: Silent Payments, which can allow you to pay someone secretly without interaction. Normally you would generate a new receive address, and send this to someone over some secure channel. But if this is not possible then there's a new BIP that uses some clever math to obscure payments.

What it means is that if I post my Taproot address here (bc1p2tz9len4slud0djlp7x9l4fhuhnjxlwmzmkd7sxfhq07rmuz8crsh97v0z), you could compute a payment address which nobody here would be able to compute, and send the funds to that, and I would be able to scan the blockchain and find this secret payment.

The clever math bit: My public key B is equal to the private key b multiplied by the generator (B = b * G). If you have public key A (with private key a) and want to send me funds, you send to an address with the public key K = hash(a * B) + B.

Now you might wonder how I would detect such an output unless I know your private key a. Well if we do the math again we get the following:

a * B = a * (b * G) // B is just the private key multiplied by g

Redistributing the terms gives:

a * B = A * b.

I know both your public key A from the transaction, and I know my private key b, so I can compute a * B (but nobody else can), so now it's simple to scan for such an output and be able to spend it. What's cool is that while I just shared my address, the secret address you compute from that address would not be known by anyone but you and I