The cubic cryptosystem is the improvement of the classical multivariable cryptosystem Square. By increasing the degree of central mapping from square mapping to cubic mapping, the public key polynomial was promoted from quadratic to cubic in order to resist the MinRank attack against the quadratic multivariable public key cryptosystem. Aiming at this system, a MinRank attack combining with difference was proposed to recover its private key. Firstly, the central mapping difference of the system was analyzed, and its rank was determined according to the structure after difference. Then, the difference of the public key was solved and the coefficient matrices of the quadratic term were extracted. After that, a MinRank problem was constructed by the coefficient matrix and the determined rank. Finally, the extended Kipnis-Shamir method was combined to solve the problem. The experimental results show that the private key of cubic cryptosystem can be recovered by using MinRank attack.