  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 M\"obius 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.
