A Survey of Structural Attacks on Post-Quantum Cryptosystems Based on Goppa Codes: Evolution, Scope, and Impact on their Variants in the Hamming Metric

Abstract

With the advent of quantum computing, public-key cryptographic systems such as RSA (Rivest–Shamir–Adleman) and ECC (Elliptic Curve Cryptography) have become vulnerable to Shor’s quantum algorithm, leading to a growing interest in post-quantum cryptography. Among the post-quantum approaches, code-based cryptography stands out for its strong theoretical foundations and proven resistance to both classical and quantum attacks. The McEliece cryptosystem is the most emblematic example. It is based on binary Goppa codes, which are algebraic block codes defined in the Hamming metric. However, it suffers from a major limitation due to the large size of its public keys, which restricts its practical applicability. To overcome this limitation, several variants of the original scheme have been introduced. Nevertheless, these new cryptographic schemes often introduce additional algebraic structures that attackers can exploit, thereby enabling new forms of structural attacks and, in some cases, weakening the robustness of the original scheme. This work presents a survey of the approaches proposed with the aim of developing structural attacks against systems based on Goppa codes. It highlights their theoretical foundations, the algorithmic methods employed, as well as their practical scope and limitations, particularly in the context of compact variants of the McEliece cryptosystem. This study also traces the evolution of these approaches toward certain variants of the McEliece scheme and analyzes how they have led to more targeted and efficient attacks. These generalizations have often made it possible to simplify or optimize earlier, more complex attacks.