Abstract: We determine the minimum positive entropy of complex Enriques surface automorphisms. This together with McMullen's work completes the determination of the minimum positive entropy of complex surface automorphisms in each class of Enriques-Kodaira classification of complex surfaces. As in McMullen's work, we finally reduce the problem to computer algebra. In this talk, after recalling known results and differences from Enriques case, I would like to explain how one can reduce this problem to finite computational problems which can be done by computer. This is a joint work with Professor Keiji Oguiso.