Alternative formula of Tm in scalar multiplication on Koblitz curve

Let E be an elliptic curve defined over F2m and T denotes the Frobenius endomorphism from the set E(F2m) to itself. The Koblitz curves are a special type of curves for which the T can be used for improving the performance of computing a scalar multiplication. In this paper, we give an alternative formula for Tm = rm + smT where rm and sm are integers. We used the triangular and pyramid numbers, Nicomachus’s theorem and Faulhaber’s formula to produce the formula for Tm. Through this approach, we improve the technique to express Tm.