McEliece Cryptosystem: Reducing the Key Size with QC-LDPC codes

Post-quantum cryptography is a growing area since Shor showed that a quantum computer with enough qubits could be used to break the most widely used public-key cryptographic protocols today, such as RSA or those based on the discrete logarithm problem. For this reason, it has become urgent to design cryptosystems that are robust against quantum computer attacks. One of them is the code-based McEliece cryptosystem, which was originally proposed using Goppa codes in 1978. The improved version of the original McEliece cryptosystem, called Classic McEliece, made it as far as the fourth round of the NIST Post-Quantum Cryptography standardization process launched by the National Institute of Technology to update the standards and include post-quantum cryptography in digital signatures, encryption and key exchange. In this work we describe and analyze two variants of the original cryptosystem designed to overcome its main drawbacks, such as its large key size and weakness against known attacks. In addition, both the recent attack that allows the recovery of the private key with limited complexity and the ways in which this attack can be prevented by changing the shape of some constituent arrays in these two new variants are discussed.