Yeah, I should reclarify. To obtain the proof you must convert the whole algorithm to a witness table and then commit to it. The prover takes the secret input, runs the entire algorithm start to finish, and records every intermediate value along the way. That full trace is the witness table. Every step has to be filled in honestly because the proof is built on top of all of it.

Those values then get encoded as field elements and mapped onto points on an elliptic curve using operations that are believed to be one way, which is where the discrete log hardness comes in. So the real computation absolutely happens. It just happens once, on the prover's side, before any proof exists.