| Summary: | Suppose E an elliptical curve defined over F2m and Ττ is Frobenius endomorphism from set
with E(F2m) to itself. Koblitz curve is a special type of curves with Ττ already being used to
improve the performance of scalar multiplication nP’s computation. P is a point that goes through
the curve. Whereas its multiplier is a non-adjacent Ττ-adic (TNAF) form whose digits are generated
by repeating division of an integer in Z(Ττ) by Ττ. Previous research has found that Ττm = Ττm + smΤτ
with integers Ττm and sm play an important role in identifying the patterns of TNAF’s expansion.
In this paper, we give a formula for coefficients aim in sm for i ≤ 6. We apply triangle’s number,
pyramid’s number, Theorem Nicomachus and Faulhaber’s formula in addition to mathematical
induction to prove this formula. With this approach, the new expression for rm for some m can
be produced to identify odd and even situations in the pseudoTNAF’s system
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