Till now geometric structures don't play a major role in cryptography. Gilbert, MacWilliams and Sloane introduced in 1974 an authentication scheme in the projective plane and showed its perfectness in the sense of the definition of Shannon. In this paper we will show that this authentication scheme also fulfills the requirement of completeness according to Kam and Davida and we will extend the application of geometric structures in cryptography by introducing an encryption scheme in the Möbius plane. We will further examine its properties, showing that it also fulfills the requirement of completeness and Shannon's requirement of perfectness in first approximation. The results of this paper can be used to define similar encryption schemes in the circle geometries of Laguerre and Minkowski.