In ECC, If We Add Two Points on the Curve, We Always Get Another Point on the Curve

Elliptic Curve Cryptography (ECC) protects your online identity and your privacy like few other things. With ECC, we take our equation of y²=x³ +7 (mod 37), and where we have valid points of: (3, 16) (4, 16) (5, 13) (6, 1) (8, 1) (9, 12) (12, 12) (13, 13) (16, 12) (17, 6) (18, 17) (19, 13) (22, 6) (23, 1) (24, 17) (30, 16) (32, 17) (35, 6) [here]. For example if x=3, we get 3³+7=34, and where 16² (mod 37) = 34. The graph is here.

With ECC, if we have a point on the curve, and then add another point, we will always get another point on the curve. It is this property that allows ECC to implement cryptography.

So, let’s try one point, and calculate the addition to it with all the other points. If (x1,y1) is not equal to (x2,y2), to two points (x1,y1) and (x2,y2), we use:

Let s=(y1−y2)/(x1−x2)

Then:

x2=s²−x1−x2

y2=s(x1−x2)−y1

If they are the same point (x1,y1), we use:

s=((3*x1²)+a) /(2*y1,p)