The study of rational point sets on circles over the Euclidean plane is discussed in a more general framework, i.e. we generalize the notion rational and consider these circular point sets over arbitrary fields. We also determine the cardinality of maximal circular point sets which depends on the radius of the corresponding circle and the characteristic of the underlying field. For the construction of them we use the so called perfect distances which have the necessary compatibility properties to find new points on a circle such that all these points still have rational distance from each other. Then we define the rotation group where its elements are the points on a circle over an arbitrary field and find a connection between a subgroup of it and perfect distances if our field is a prime field. Furthermore, we describe a possible application in cryptography of the rotation group similar to the Diffie-Hellman key exchange.