Post-quantum cryptography based on codes: State of the art and open challenges

A new generation of cryptographic primitives is on the horizon and they are likely to replace soon many previous and widespread systems. This is due to the cybersecurity threat represented by the appearance of the first practical quantum computers, that will be able to easily solve several mathematical problems on which classical cryptographic systems rely. Cryptographic primitives based on error correcting codes are among the most promising candidates for this new generation of post-quantum algorithms that will be established in the near future. This paper provides an up-to-date overview of the features and performances of these systems. We review classical Goppa code-based systems and assess their security taking into account quantum speedups of the most efficient attacks. We also compare them with promising alternatives based on QC-LDPC and QC-MDPC codes. We then consider another family of codes known as monomial codes and assess their usability in the same framework.