The modern security protocols in most of our systems rely primarily on three basic functions of asymmetric cryptography: public key encryption, digital signature, and key exchange. Today, we only do key exchange (TLS 1.3) with the ECDH protocol. The confidentiality is persistent because the session keys are discarded at the end and to certify this key exchange, we sign it with RSA or ECDSA. However, these cryptosystems are at least theoretically attackable in a quantum computer model. Thus the NIST PQC standardization process has given significant momentum to research on code-based public-key cryptosystems specifically. Their security is based on the hardness of the syndrome decoding problem. In this article, we first propose a new formalism of the matrix-vector product in based-code cryptography. Second, we present a chosen-ciphertext attack on the first step of Niederreiter decryption by solving the matrix-vector product problem with side-channel information. Finally, we put this result to recover secret information in code-based cryptosystems including some candidates for the extension of the NIST PQC normalization process.