F.  Yunos, S.M.  Suberi, Sh.K.  Said Husain, M.R.K  Ariffin, M.A.  Asbullah, 
    On Some Specific Patterns of τ-Adic Non-Adjacent Form Expansion over Ring Z (τ),
    Journal of Engineering and Applied Sciences,
    Volume 14,Issue 23,
    2019,
    Pages 8609-8615,
    ISSN 1816-949x,
    jeasci.2019.8609.8615,
    (https://makhillpublications.co/view-article.php?doi=jeasci.2019.8609.8615)
    Abstract: Let &#964;=(-1)<sup>1-a</sup>+&#8730;-7/2 for a&#8712;{0, 1} is Frobenius map from the set E<sub>a</sub>(F<sub>2</sub>m) to it self for a point (x, y) on 
Koblitz curves E<sub>a</sub>. Let P and Q be two points on this curves. &#964;-adic Non-Adjacent Form (TNAF) of &alpha; an element of the ring Z(&#964;) = {&alpha; = c+d&#964;|c, d&#8712;Z} is an expansion where the digits are generated by successively dividing
&alpha; by &#964;, 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, c<sub>1</sub>, &#133;, c<sub>1-1</sub>], [-1, c<sub>1</sub>, &#133;, c<sub>1-1</sub>], [1, c<sub>1</sub>, &#133;, c<sub>1-1</sub>] and [0, 0, 0, c<sub>3</sub>, c<sub>4</sub>, &#133;, c<sub>1-1</sub>].
    Keywords: element;expansion;Frobenius map;successively;&#964;-adic non-adjacent form;Koblitz curve;TNAF