Chris Busenhart | Searcharxiv

arXiv2024

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

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.

math.CO↗

arXiv2024

Geometric Aspects to Diophantine Equations of the Form $x^2 + zxy + y^2 = M$ and $z$-Rings

In the following we consider Diophantine equations of the form $x^2+ zxy + y^2 = M$ for given $M,z \in \mathbb{Z}$ and discuss the number of its (primitive) solutions as well as the construction of them. To reach this goal we introduce $z$-rings which turn out to be a useful tool to investigate these Diophantine equations. Moreover, we will extend these rings and study the algebraic curves defined by them on a plane by methods inspired by the complex plane. Then we define the so called subbranches which are bounded and connected parts of the algebraic curves containing a representative of each solution of the Diophantine equations with respect to association in $z$-rings. With the help of them we can easily prove the existence or non-existence of solutions to the above Diophantine equations. Then we divide the integer primes with respect to the different $z$-rings into two main categories, i.e. the regular and irregular elements. We show that the irregular elements are prime in the corresponding $z$-rings and we identify that most of the $z$-rings cannot be unique factorization domains. We determine the number of positive, primitive solutions of the above Diophantine equation if $M \in \mathbb{N}$ is a product of irregular elements in the corresponding $z$-ring for $z \in \mathbb{N}$. We also give an overview how many primitive and non-primitive solutions in a given quadrant we can find for arbitrary $M,z \in \mathbb{Z}$, especially, if $M$ is a power of any irregular element. Furthermore, we consider the case $z = 3$, determine the regular and irregular elements as well as the number of positive, primitive solutions of the Diophantine equation $x^2 + 3xy + y^2 = M$ depending on $M \in \mathbb{N}$.

math.NT↗