On some specific patterns of τ-adic non-adjacent form expansion over ring Z(τ): An alternative formula

Abstract

Let τ=(−1)1−a+−72 for a e {0,1} is Frobenius map from the set Ea (F2m) to itself 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 paper, we find the alternative formulas for TNAF that have specific patterns [c0, 0, …, 0, cl-1], [c0, 0, …, Cl−12, …, 0, cl-1], [0, c1, …, cl-1], [−1, c1, …, cl-1] and [0,0,0,c3, c4, …, cl-1] by applying τm = −2sm-1 + sm τ for sm=∑i=1| m+12 |(−2)i−1tm+1(i−1)!∏j=12i−2(m−j).