Cryptographic systems refer to a set of algorithms needed to implement a particular form of encryption, decryption, signature and verification, which have practical applications in engineering area. Among cryptographic systems, Multivariate Public Key Cryptography (MPKC) is one of the most popular post-quantum candidates since it has the potential to resist quantum computer attacks. MPKC must be protected against a wide range of attacks, including side-channel attack, which is any attack based on information gained from the physical implementation of cryptographic systems. However, there are few side-channel attacks on MPKC schemes. In this paper, we present techniques to exploit Differential Power Analysis and fault analysis attacks for analyzing the effectiveness of side-channel attacks on MPKC schemes. We propose a general model of side-channel attacks on enhanced Tame Transformation Signature (enTTS) scheme, which is one of the representative MPKC schemes. We implement a naive enTTS scheme on Application Specific Integrated Circuits and propose a successful side-channel attack on the implementation. Experimental results show that our attack successfully obtains all the pieces from the private keys of the enTTS scheme and they clearly demonstrate the importance of protecting MPKC against side-channel attacks.