On some specific patterns of τ-Adic non-adjacent form expansion over ring Z (τ)

Let τ=(-1)1-a+√-7/2 for a∈{0, 1} is Frobenius map from the set Ea(F2m) to it self for a point (x, y) on Koblitz curves Ea. Let P and Q be two points on this curves. τ-adic Non-Adjacent Form (TNAF) of α an element of the ring Z(τ) = {α = c+dτ|c, d∈Z} is an expansion where the digits are generated by successively dividing α by τ, allowing remainders of -1, 0 or 1. The implementation of TNAF as the multiplier of scalar multiplication nP = Q is one of the technique in elliptical curve cryptography. In this study, we find the formulas for TNAF that have specific patterns [0, c1, …, c1-1], [-1, c1, …, c1-1], [1, c1, …, c1-1] and [0, 0, 0, c3, c4, …, c1-1].

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