We study the stability of finite difference schemes for hyperbolic initial boundary value problems in one space dimension. Assuming stability for the dicretization of the hyperbolic operator as well as a geometric regularity condition, we show that an appropriate determinant condition, that is the analogue of the uniform Kreiss-Lopatinskii condition for the continuous problem, yields strong stability for the discretized initial boundary value problem. The analysis relies on a suitable discrete block structure condition and the construction of suitable symmetrizers. Our work extends the results of Gustafsson, Kreiss, Sundstrom to a wider class of finite difference schemes.