Decoding algorithms for Goppa codes with errors and erasures

In 1978, McEliece built the first public key cryptosystem based on error-correcting codes. This cryptosystem based on Goppa codes is considered promising and cryptographically stable, taking into account quantum computing. At the same time, effective attacks on the secret keys of this cryptosystem have not yet been found. In the paper, algorithms for decoding Goppa codes in the case of errors and erasures are investigated. Four decoding algorithms based on the algorithms for Reed–Solomon codes proposed by Gao, Berlekamp and Massey, Sugiyama, and others are given. The first two algorithms are based on Gao algorithm and related to syndrome-free decoding algorithms, the rest are related to syndrome decoding algorithms. Moreover, any of these algorithms is also applicable for the case of a communication channel with errors only. Examples of decoding separable Goppa codes using these algorithms are also given.