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Till now geometric structures don't play a major role in cryptography.Gilbert, MacWilliams and Sloane introduced in 1974 an authentication scheme inthe projective plane and showed its perfectness in the sense of the definitionof Shannon. In this paper we will show that this authentication scheme alsofulfills the requirement of completeness according to Kam and Davida and wewill extend the application of geometric structures in cryptography byintroducing an encryption scheme in the M\"obius plane. We will further examineits properties, showing that it also fulfills the requirement of completenessand Shannon's requirement of perfectness in first approximation. The results ofthis paper can be used to define similar encryption schemes in the circlegeometries of Laguerre and Minkowski.

Design of Ciphers based on the Geometric Structure of the M\"obius Plane