BASIC SCHEME

The recipient publishes their silent payment address, a single 32 byte public key:

X = x*G

The sender picks an input containing a public key:

I = i*G

The sender tweaks the silent payment address with the public key of their input:

X' = hash(i*X)*G + X

Since i*X == x*I (Diffie-Hellman Key Exchange), the recipient can detect the payment by calculating hash(x*I)*G + X for each input key I in the blockchain and seeing if it matches an output in the corresponding transaction.