What does Pay to Contract (P2C) mean?

Question

What does Pay to Contract (P2C) mean in Bitcoin? Is it just tweaking private/public key with the hash (represented as a number) of some message and then using classic P2PKH?

For example, if we are buying some car, the message this payment is for the car goes into SHA256 hash function. After that, tweaking the given number with the public key and by making RIPEMD(SHA256()) hash of it, we create classic P2PKH output. Later, we can prove that the payment was for the car with revealing the original public key and the message.

Is it P2C? Does it represent something more?

Answer 1

The actual protocol as described all the way back in 2012 in the paper Homomorphic Payment Addresses and the Pay-to-Contract Protocol is a bit more complicated, but the basics are as you've described them: create a hash from a message, then tweak the public key with this hash and you'll be able to prove later that the public key commits to your original message. Taproot implements a version of this protocol, as spending using the script path requires you to prove that your public key commits to the root of a script tree.

Bitcoin Optech has a topic for Pay-to-Contract which summarizes the topic and links to a few instances where the protocol is discussed.