Analysis of New Variants of the McEliece Cryptosystem Based on Convolutional Codes

• Autores: Miguel Beltrá Vidal
• Directores de la Tesis: Diego Napp Avelli (dir. tes.), Paulo José Fernandes Almeida (dir. tes.)
• Lectura: En la Universitat d'Alacant / Universidad de Alicante ( España ) en 2025
• Idioma: español
• Número de páginas: 171
• Tribunal Calificador de la Tesis: Joachim Rosenthal (presid.), Verónica Requena Arévalo (secret.), F.J. Lobillo Borrero (voc.)
• Programa de doctorado: Programa de Doctorado en Ciencias Experimentales y Biosanitarias por la Universidad de Alicante
• Materias:
  • Matemáticas
    • Ciencia de los ordenadores
      • Código y sistemas de codificación
• Enlaces
  • Tesis en acceso abierto en: RUA
• Resumen

  • It has been known since the mid-1990s that building a sufficiently large quantum computer would threaten the vast majority of public-key cryptosystems currently in use. Recent advances in quantum computing have led to a need to design and standardize public-key cryptosystems that are secure against quantum computers. Among the candidate cryptosystems is the McEliece cryptosystem. This cryptosystem was published in 1978 by mathematician Robert J. McEliece and bases its security on NP-complete problems in coding theory. The main disadvantage of this cryptosystem is the large size of its public key. For this reason, other alternatives have been preferred in practice. However, after almost five decades, the cryptosystem and its dual version, the Niederreiter cryptosystem, remain secure, and no efficient attacks against it are known, either with classical or quantum computers. For this reason, it is considered one of the leading candidates for post-quantum cryptography. This has motivated researchers to design variants of this cryptosystem that require a smaller public key than the original version. This thesis falls precisely within this line of research. It studies a series of variants of the McEliece cryptosystem based on convolutional codes. The idea of these variants is to mask an efficiently decodable linear code with some polynomial transformations. In the McEliece setting, the public key is the encoder of a convolutional code and the ciphertext is a sequence of vectors with some errors intentionally added to distort the message. Decryption consists in correcting these errors. In the Niederreiter setting, the public key is the parity-check matrix of a convolutional code and the ciphertext is the sequence of syndromes of some sequence of low weight vectors. Decryption consists of retrieving these vectors using the syndrome decoding algorithm of the secret code. For both schemes, we study two versions: one with a binary Goppa code as the secret code and one with a Generalized Reed-Solomon code, and show how to construct the corresponding polynomial transformations in each case. Moreover, we study how to construct the Niederreiter scheme in a particular way to achieve systematic public keys, which results in smaller keys than in the non-systematic case. These variants are new to the literature. It is shown that with the adopted approach, it is possible to achieve a reduction in the size of the public key to 16.74% of the original key size, that is, approximately one-sixth, without sacrificing the security of the cryptosystem.