A large part in the design of secure cryptographic primitives consists in identifying hard algorithmic problems. Despite the fact that several problems have been proposed as a foundation for public-key primitives, those effectively used are essentially classical problems coming from integer factorization and discrete logarithm. On the other hand, coding theory appeared with the goal to solve the challenging problem of decoding a random linear code. It is widely admitted as a hard problem that has led McEliece in 1978 to propose the first code-based public-key encryption scheme. The key concept is to focus on codes that come up with an efficient decoding algorithm. He also advocated the use of binary Goppa codes. Since then, it belongs to the very few cryptosystems which remain unbroken. This thesis is primarily interested in studying the security of code-based primitives. The first category we analyzed consists of variants of the McEliece cryptosystem. Our works expose practical key-recovery attacks either by mounting dedicated techniques, or by devising algebraic attacks. This latter result also provides a new framework to assess the security of the McEliece cryptosystem and a first step towards the design of attacks based on the solving of algebraic systems. Furthermore, we show that this approach can be used to study the Goppa Code Distinguishing problem, which asks whether there is an efficient way to distinguish a Goppa code from a randomly drawn linear code. It represents an important assumption which supports the use of Goppa codes in cryptography. We show that it is possible to efficiently solve it as long as the code rate is sufficiently high. Finally, we investigate the security of a signature scheme based on two random linear codes. Our analysis shows that the attack is sensitive to their rates and can be practical when the rates are close.