Topics in Algebra: Elliptic curves
Question
Given the elliptic curve E: y2 = x3 – 3x + 1, use the group law to compute 2P and 3P for a
point P = (x1, y1)
Solution
To compute 2P = P + P, the formula for the point doubling is;
λ = 3x12−3 /2 y 1
P = (x1, y1) is a point on the curve.
The coordinates of 2P = (x2, y2) are;
x2 = λ2 − 2x1
y2 = λ (x1 − x2) − y1
2P = (x2, y2), 3P = P + 2P
Adding two points using the formula;
y 2− y 1
x 2−x 1
Points in the eclipse curve
2P = (x2, y2) and Q = (x2, y2)
y 2− y 1
λ=
x 2−x 1
where P ≠ QP, and the coordinates of 3P = (x3, y3) are:
x3 = λ2 − x1 − x2
y3 = λ (x1 − x3) − y1
Therefore, Coordinates of 3P = (λ2 − x1 − x2, λ (x1 − x3) − y1)
Question
Given the elliptic curve E: y2 = x3 – 3x + 1, use the group law to compute 2P and 3P for a
point P = (x1, y1)
Solution
To compute 2P = P + P, the formula for the point doubling is;
λ = 3x12−3 /2 y 1
P = (x1, y1) is a point on the curve.
The coordinates of 2P = (x2, y2) are;
x2 = λ2 − 2x1
y2 = λ (x1 − x2) − y1
2P = (x2, y2), 3P = P + 2P
Adding two points using the formula;
y 2− y 1
x 2−x 1
Points in the eclipse curve
2P = (x2, y2) and Q = (x2, y2)
y 2− y 1
λ=
x 2−x 1
where P ≠ QP, and the coordinates of 3P = (x3, y3) are:
x3 = λ2 − x1 − x2
y3 = λ (x1 − x3) − y1
Therefore, Coordinates of 3P = (λ2 − x1 − x2, λ (x1 − x3) − y1)
