Maximal Circular Point Sets over Arbitrary Fields and an Application to Cryptography

TL;DR

This paper generalizes the concept of rational point sets on circles to arbitrary fields, determines their maximal sizes based on field characteristics, and explores cryptographic applications using rotation groups.

Contribution

It introduces a framework for circular point sets over arbitrary fields, characterizes their maximal sizes, and links these structures to cryptographic protocols.

Findings

Maximal circular point set sizes depend on field characteristic and circle radius.

Established a connection between rotation groups and perfect distances over prime fields.

Proposed a cryptographic application analogous to Diffie-Hellman key exchange.

Abstract

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.