Triangle Conics, Cubics and Possible Applications in Cryptography

Veronika Starodub,Sergii S Podpriatov,Ruslan V Skuratovskii

doi:10.13189/ms.2021.090515

Abstract

We research triangle cubics and conics in classical geometry with elements of projective geometry. In recent years, N.J. Wildberger has actively dealt with this topic using an algebraic perspective. Triangle conics were also studied in detail by H.M. Cundy and C.F. Parry recently. The main task of the article is development of a method for creating curves that pass through triangle centers. During the research, it was noticed that some different triangle centers in distinct triangles coincide. The simplest example: an incenter in a base triangle is an orthocenter in an excentral triangle. This is the key for creating an algorithm. Indeed, we can match points belonging to one curve (base curve) with other points of another triangle. Therefore, we get a new fascinating geometrical object. During the research number of new triangle conics and cubics are derived, their properties in Euclidian space are considered. In addition, it is discussed corollaries of the obtained theorems in projective geometry, which proves that all of the discovered results could be transferred to the projective plane. It is well known that many modern cryptosystems can be naturally transformed into elliptic curves. We investigate the class of curves applicable in cryptography.

Highlights

Centers of triangle and central triangles were studied by Clark Kimberling [1]
Geometry of conic sections and other triangle curves are broadly used in the projective geometry we looked on the obtained result through the prism of the projective geometry
As was shown in the introduction part, each curve and assigned to it triangle centers may be applied in cryptography as a set of secret keys

Summary

Introduction

Centers of triangle and central triangles were studied by Clark Kimberling [1]. We considered number of curves that pass through base triangle centers, as incenter, othocenter, circumcenter, mittenpunkt, Bevan point, and others. We provide an approach to construct Edwards curves of determined order, which are important within the cryptography and coding theory domains. We note, that it was accepted in 1999 as an ANSI standard and in 2000 as an IEEE and NIST standard. In this work we try to improve ECDSA (Elliptic Curve Digital Signature Algorithm) by coding ellipitc curve in terms of private geometrical parameters. This parameters can be used as components of private kee. Mathematical part of this investigation belongs to V.

Preliminaries

Main result

Discussions

Conclusions