TY  - JOUR
T1  - On Some Specific Patterns of τ-Adic Non-Adjacent Form Expansion over Ring Z (τ)
AU - Yunos, F. AU - Suberi, S.M. AU - Said Husain, Sh.K. AU - Ariffin, M.R.K AU - Asbullah, M.A. 
JO  - Journal of Engineering and Applied Sciences
VL  - 14
IS  - 23
SP  - 8609
EP  - 8615
PY  - 2019
DA  - 2001/08/19
SN  - 1816-949x
DO  - jeasci.2019.8609.8615
UR  - https://makhillpublications.co/view-article.php?doi=jeasci.2019.8609.8615
KW  - element
KW  -expansion
KW  -Frobenius map
KW  -successively
KW  -τ-adic non-adjacent form
KW  -Koblitz curve
KW  -TNAF
AB  - 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>].
ER  -