Constructing Scalar Multiplication via Elliptic Net of Rank Two

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.

Scalar Multiplication via Elliptic Net using Generalized Equivalent Sequences

“…Meanwhile, Stange implemented her net algorithm to calculate pairings [6] and Stange's algorithm was adapted to calculate elliptic curve scalar multiplication by Kanayama et al [7]. The earlier discussion of the elliptic net by Malaysian researchers can be seen in Muslim & Said [8]- [9].…”