Abstract
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.
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