Multivariate Public Key Cryptography (MPKC) is one of the promising candidates of Post-quantum Cryptography (PQC) that ensures data security, information security, computer system security, and network security against large-scale quantum computers. Our research work in this article focuses on analyzing rank weakness in cubic cryptographic structures. The extension of the rank attack from quadratic to cubic polynomial system demonstrates the occurrence of the same rank fault as in the case of multivariate quadratic polynomial-based cryptosystems. In this paper, we present the cryptanalysis of the multivariate cubic polynomial-based digital signature algorithm, which is basically based on the big field system called Hidden Field Equation (HFE) and uses the structure of the state-of-the-art signature scheme in multivariate cryptography called Rainbow. In addition to this, we provide a countermeasure to overcome the presented attack on the MPKC digital signature.