This paper is concerned with the computation of Nash solutions for a class of deterministic linear-quadratic two-person nonzerosum difference games. The dynamics of the system is characterized by a linear discretetime descriptor equation Ex_k+1=Ax_k+Bu_k+Cv_k, where E is singular. The games are studied under the open-loop information structure. It is proven that the linear quadratic Nash game of the discrete descriptor system always admit uncountable Nash equilibrium solutions even if the information structure of the game is open-loop. This property is very different from that in the open-loop Nash game of a state space system. A pair of minimum-norm open-loop Nash solutions is computed by recursively solving two coupled asymmetric Riccati-type matrix equations.