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

Chris Busenhart

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.

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

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.

Paper Structure

This paper contains 11 sections, 21 theorems, 97 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction and framework
  2. Basic notions and tools
  3. Definitions and facts
  4. Parametrization of points on circles and consequences
  5. Maximal circular point sets
  6. The case if $\mathbb{F}$ is a prime field
  7. Perfect distances
  8. C-maximal circular point sets on circles over arbitrary field planes
  9. Application in Cryptography
  10. Rotation groups on circles
  11. A possible application in cryptography

Key Result

Lemma 2.8

Let $\mathbb{F}$ be an arbitrary field and $P,Q \in \mathbb{F}^2$. Then the squared distance of $P,Q$ remains invariant under translations and rotations.

Figures (4)

  • Figure 1: The points of $C \left((0,0),1 \right)_{\mathbb{F}_7}$ with rational squared distances (blue) and non-rational squared distances (red)
  • Figure 2: The points of $C \left((7,11),6 \right)_{\mathbb{F}_{13}}$ and their rational squared distances
  • Figure 3: The points of $C \left((0,0),1 \right)_{\mathbb{F}_{49}}$ and the squared distances between them and the point $\left( a + 4,5a + 2 \right)$ marked by red lines for non-rational and blue lines for rational squared distances
  • Figure 4: Three points $A,B,C$ on a circle $C \left(M,r \right)_{\mathbb{F}}$ over a prime field plane $\mathbb{F}$ with rational squared distances denoted by $a, b$ and squared distance $c$.

Theorems & Definitions (66)

  • Definition 2.4
  • Definition 2.5
  • Example 2.6
  • Definition 2.7
  • Lemma 2.8
  • proof
  • Definition 2.9
  • Example 2.10
  • Definition 2.11
  • Definition 2.12
  • ...and 56 more