On the $s$-projected Goppa Codes and their Application to the Interactive Oracle Proofs of Proximity

Error-correcting codes are a useful tool in cryptography, being used both to design schemes whose hardness relies on coding theory problems (the area typically known as code-based cryptography) but also as a building block in other contexts, such as proof systems and associated cryptographic argument systems. Among all codes used in cryptography, Goppa codes stand out, notably in the well-known McEliece encryption scheme, where they have withstood the test of time as one of very few classes of codes to be considered secure, for over 40 years. Therefore, a natural question arises whether Goppa codes can also be used in the construction of proof systems or cryptographic argument systems. In the context of probabilistically checkable proofs (PCPs), codes are used to transform proofs into a format that requires only a constant number of queries for verification. This efficiency is often achieved using low-degree polynomial codes, which enable the verifier to check the consistency through interpolation, ensuring correctness with minimal effort. This application underscores the importance of good codes in enhancing the robustness and verifiability of proof systems. Given their critical role, we take a closer look at how error-correcting codes are used in proof systems to find a new way of obtaining an interactive oracle proof system of proximity (IOPP) from Goppa codes.