Multivariate Cryptography between Algebra and Applications

This thesis investigates multivariate cryptography from both an algebraic and applied perspective, with a focus on its role in post-quantum cryptography. The work studies multivariate quadratic polynomial systems over finite fields, emphasizing Oil and Vinegar structures, mixed systems, and the algebraic invariants that influence their cryptographic security. Particular attention is devoted to Hilbert functions, Hilbert series, degree of regularity, and first fall degree, which are used to estimate the complexity of solving such systems through algebraic techniques such as Gröbner basis algorithms. The thesis derives structural results for homogeneous Oil and Vinegar and mixed polynomial systems, providing tools to evaluate their resistance against known algebraic attacks. These theoretical contributions are then applied to the design and analysis of OliVier, a cryptographic construction based on overdetermined mixed systems, with discussion of its security motivations, parameter choices, and decryption efficiency. Finally, the thesis explores possible future directions, including the use of symmetric polynomials in multivariate digital signature schemes. Overall, the work contributes to the understanding of the algebraic foundations of multivariate cryptography and their potential applications in the development of quantum-resistant cryptographic protocols.