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QUAD stream cipher uses multivariate polynomial systems. It has provable security based on the computational hardness assumption. More specifically, the security of QUAD depends on hardness of solving non-linear multivariate system us over a finite field, and it is known as an NP-Hard problem. However, QUAD is slower than other stream ciphers, and an efficient implementation, which has a reduced computational cost is required. In this paper, we propose an efficient implementation of computing multivariate polynomial systems for multivariate cryptography on GPU and evaluate efficiency of the proposal. GPU is considered to be a commodity parallel arithmetic unit. Moreover, we give an evaluation of our proposal. Our proposal parallelizes an algorithm of multivariate cryptography, and makes it efficient by optimizing the algorithm with GPU.