One promising post-quantum security candidate is Multivariate Public Key Cryptosystems (MPKCs); they are susceptible to algebraic attacks. In this paper, I will discuss the vulnerability of MPKCs to higher-order algebraic cryptanalysis, which transforms nonlinear systems of polynomies into structured ideals on finite fields. The aim is to assess the efficiency of the attacks through the method of computation of Gröbner basis, linearization and re-linearization. The algorithm is a combination of Buchberger and F4/F5 algorithms, and is supported by experimental study of benchmark MQ instances. The comparative evaluation reports a key recovery success rate of 94.2% for the proposed algebraic attack, compared with 62.5% for classical brute-force and 79.8% for hybrid cryptanalysis. The corresponding computation times are 128, 356, and 214 seconds, respectively, representing reductions of 64.04% and 40.19% relative to the two baselines. These preliminary results indicate improved attack efficiency under the evaluated conditions; however, further benchmark specification and statistical validation are required to establish reproducibility. The findings highlight the vulnerabilities in the structure of specific parameter settings and the safe design suggestions. The paper contributes a new and meticulous evaluation framework to the existing knowledge base on the notion of algebraic resilience in post-quantum cryptography.