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A τ-adic non-adjacent form (TNAF) of an element α of the ring Z(τ) is an expansion whereby the digits are generated by iteratively dividing α by τ, allowing the remainders of -1,0 or 1. The application of TNAF as a multiplier of scalar multiplication (SM) on the Koblitz curve plays a key role in Elliptical Curve Cryptography (ECC). There are several patterns of TNAF (α) expansion in the form of [c0,0,…,0,cl-1 ], [c0,0,…,c(l-1)/2,…,0,c(l-1)], 2+2k, 3+4k, 5+4k and 8k1+8k2 that have been produced in prior work in the literature. However, the construction of their properties based upon pyramid number formulas such as Nichomacus’s theorem and Faulhaber’s formula remains to be rather complex. In this work, we derive such types of TNAF in a more concise manner by applying the power of Frobenius map (τm) based on v-simplex and arithmetic sequences.

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.