What is the difference between a deterministic and interactive proof system?

The difference between a deterministic and interactive proof system lies in the level of interaction between the prover and the verifier.

In a deterministic proof system, the prover provides a fixed proof that can be verified by the verifier without any further interaction. The prover's proof is a complete and deterministic solution that convinces the verifier of the correctness of a statement or problem. The verifier simply checks the proof and accepts or rejects it based on its validity.

On the other hand, an interactive proof system involves a more dynamic and interactive process between the prover and the verifier. Here, the prover and verifier engage in a series of back-and-forth interactions, where the prover provides partial information or evidence, and the verifier asks questions or requests additional information. This interactive process continues until the verifier is convinced of the correctness of the statement or problem.

The key distinction is that in an interactive proof system, the verifier's decision is based not only on the proof provided by the prover but also on the interactions and responses during the protocol. This allows for a more flexible and efficient verification process, as the verifier can adapt its queries based on the prover's responses.

Interactive proof systems are particularly useful in scenarios where the problem being verified is computationally complex or difficult to solve deterministically. By engaging in an interactive process, the prover can convince the verifier of the solution's correctness without explicitly revealing the entire solution, thus saving computational resources.

In summary, the main difference between a deterministic and interactive proof system is the level of interaction between the prover and the verifier. Deterministic proof systems involve a fixed proof that can be verified without further interaction, while interactive proof systems involve dynamic interactions between the prover and verifier to establish the correctness of a statement or problem.