Constructing Scalar Multiplication via Elliptic Net of Rank Two

Norliana Muslim,Mohamad Rushdan Md Said

doi:10.14419/ijet.v7i4.34.26884

Abstract

Elliptic nets are a powerful method for computing cryptographic pairings. The theory of rank one nets relies on the sequences of elliptic divisibility, sets of division polynomials, arithmetic upon Weierstrass curves, as well as double and double-add properties. However, the usage of rank two elliptic nets for computing scalar multiplications in Koblitz curves have yet to be reported. Hence, this study entailed investigations into the generation of point additions and duplication of elliptic net scalar multiplications from two given points on the Koblitz curve. Evidently, the new net had restricted initial values and different arithmetic properties. As such, these findings were a starting point for the generation of higher-ranked elliptic net scalar multiplications with curve transformations. Furthermore, using three distinct points on the Koblitz curves, similar methods can be applied on these curves.

Highlights

IntroductionThe term “elliptic net” and its concepts were the brainchild of [9]
The term “elliptic net” and its concepts were the brainchild of [9]. Such nets are being extensively utilized for pair-based cryptographical optimizations. With their origin from non-linear recurrence relations, first-ranked nets are known as elliptic divisibility sequences [11] apart from having addition and duplication properties [7]
The latter has been extensively studied for its additional structure that allowed speed-ups in the computations of elliptic curve scalar multiplications by means of implementing sequence of addition and duplication of points

Summary

Introduction

The term “elliptic net” and its concepts were the brainchild of [9] Such nets are being extensively utilized for pair-based cryptographical optimizations. Stange described general elliptic nets as the mapping of finite-ranked abelian groups onto integral-related domains (i.e. R). In the attempt to prove the possibility of developing elliptic nets from a new cryptographic curve, Koblitz curve had been utilized The latter has been extensively studied for its additional structure that allowed speed-ups in the computations of elliptic curve scalar multiplications by means of implementing sequence of addition and duplication of points. This study was intended to verify the relationships between elliptic functions, net polynomials, and Koblitz curves. In the final section of this paper, the study outcomes were concluded

Weierstrass and Koblitz Curve

Novel Elliptic Net upon Koblitz ρ

Conclusion