TRIAD ANALYSIS BY DESIGN, SECURITY, AND EVOLUTION OF HFE, MI,AND OIL-VINEGAR CRYPTOSYSTEMS

Abstract

 A well-known contender in the field of post-quantum cryptography is Multivariate Public Key Cryptography (MPKC), which provides security based on the computational difficulty of solving systems of multivariate quadratic equations over finite fields. The foundational trinity of the suggested schemes the Hidden Field Equations (HFE), Matsumoto-Imai (MI), and Oil-Vinegar (OV) cryptosystems has greatly influenced the development of MPKC. These three cryptosystems are reviewed in this work in a unified and comparative manner, with an analysis of their algebraic structures, key generation mechanisms, encryption and signature procedures, and known cryptanalytic attacks. The study clarifies each scheme's advantages, disadvantages, and changes over time by following its historical evolution and emphasizing its fundamental design ideas. Their contribution to standardization initiatives and their ability to withstand new quantum threats are highlighted in particular. In order to facilitate future research and optimization in safe, effective, and scalable multivariate cryptosystems for the post-quantum era, the paper attempts to offer a comprehensive knowledge.