I am new in Elliptic curve, so I started with implementing (single scalar multiplication) I have done it the simple way, and then I moved to Double & Add algorithm later with NAF form.
When I moved to the NAF form. Two versions of algorithm gave me two different results. And I'm not sure what did I miss.
I'm using ECC defined as: $y^2=x^3 - 2x +2$ over the finite field $GF(23)$
I'm trying to compute $7P$ where $P=[5,5]$, with simple double and add the result is $7P=[5,18]$ but with NAF $7P=[5,-5]$
I believe the reason is I compute $7P$ in double & add with NAF as $7P = 8P - P = O - [5,5] = [5,-5]$ so I'm not sure if I miss some important condition when working in NAF form.
Computed points:
1P: [5, 5]
2P: [15, 14]
3P: [16, 15]
4P: [11, 0]
5P: [16, 8]
6P: [15, 9]
7P: [5, 18]
8P: O
9P: [5, 5]
10P: [15, 14]
11P: [16, 15]
12P: [11, 0]
13P: [16, 8]
14P: [15, 9]
15P: [5, 18]
16P: O
double & add $7P = 1P + 2P + 4P = [5,18]$
double & add with NAF $7P = 8P - P = O - [5,5] = [5,-5]$