Multivariate Polynomial Cryptography (MPC) has emerged as a promising candidate for securing digital communication in the post-quantum era. Despite its potential, existing MPC schemes exhibit vulnerabilities that compromise their effectiveness against both classical and quantum attacks. This paper leverages discrete mathematics to address these vulnerabilities, providing a robust mathematical foundation for enhancing the security and efficiency of MPC schemes. The paper begins by identifying critical weaknesses in current MPC algorithms, such as susceptibility to algebraic attacks and high computational overhead. Using tools from discrete mathematics, particularly in combinatorics and algebra, novel multivariate quadratic equations are proposed, designed to be computationally infeasible for quantum algorithms to solve. The approach involves constructing new cryptographic primitives that improve resistance to known attack vectors while maintaining practical performance levels. This research fills a significant gap in the field by providing a rigorous mathematical framework for the development of more secure MPC schemes. Comprehensive security proofs and performance analyses demonstrate that these enhanced MPC constructions offer a viable path forward for post-quantum cryptography. The findings underscore the critical role of discrete mathematics in advancing the field and ensuring the future security of digital communications.