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Adding points on the curve applies the chord-and-tangent rule and outcomes another point. Adding points on the curve generates an additive group including the point at infinity in symbol O [4]. It is the group of points which are utilized in building cryptosystems using the curve. The description in geometric gives easy understanding. Consider that M = (x₁, y₁) and N = (x₂, y₂) are the points existing on the curve. Suppose that adding M and N results P = (x₃, y₃) on the curve. Adding points is described in Figure 4.2. The tangent line sketching from M to N touches at the point marked by −P. The reflection of −P is P [1, 6]. Consider that doubling a point M results P = (x₃, y₃) on the curve. Doubling a point is described in Figure 4.3. The tangent line sketching from point M touches at the point marked by −P. The reflection of −P is P as in the case of adding points [1, 6].

Figure 4.2 Adding points (P = M + N).

Figure 4.3 Doubling a point (P = M + M).

The algebraic methods for adding points and doubling a point on E(GF(p)) are followings [4].

 • M + O = O + M = M and M + (−M) = O for any point, M ∈ E(GF(p)). If M = (x, y) ∈ E(GF(p)), then (x, −y) is defined as (−M) which stands for the inverse of M. O is the point at infinity which is known as additive identity.

 • (Adding points). Consider that M, N ∈ E(GF(p)), M = (x1, y1), and N = (x2, y2) on the condition: M ≠ ±N. Then, M + N = (x3, y3) in which x3 = λ2 − x1 – x2, y3 = λ(x1 – x3) – y1, and λ(y2 – y1) / (x2 – x1).

 • (Doubling a point). Consider that M = (x1, y1) ∈ E(GF(p)) on the condition: M ≠ −M. Then, 2M = (x3, y3) in which x3 = λ2 – 2x1, y3 = λ(x1 – x3) – y1, and λ = (3x12 + a)2y1.