Adding Points - 2P, 3P, 4P, etc.
Based on the curve y^2 = x^3 - 2x + 4, the tangent line to the point P is equivalent to a "double root" of a cubic, which then crosses the curve at R. We take -R = 2P, and find the line that passes through P & R, which then crosses the curve at the point S. We take -S = P + R, and find the line that passes through P & S, which then crosses the curve at the point T. etc. This effectively "adds" P to the previous result each time.
Adding a point P to itself repeatedly on an elliptic curve to get P, 2P, 3P, etc., involves both geometric intuition and the algebraic group law defined for the curve. This process is foundational in the arithmetic of elliptic curves. Here’s a step-by-step description of how this addition works:
Doubling P to get 2P
1. Tangent Line: Draw a tangent line to the curve at P. This line represents the "addition" of P to itself, geometrically realised by the slope of the curve at P.
2. Find Intersection: The tangent line will intersect the curve at another point. This intersection represents P + P = 2P in the group law, but geometrically, you've found the point -2P, which is the reflection of 2P across the x-axis.
3. Reflect Across the X-Axis: Reflect the intersection point across the x-axis to find 2P, the result of adding P to itself.
Adding 2P to P to Get 3P
1. Line Through P and 2P: Draw a straight line through P and 2P. This line represents the addition of P and 2P.
2. Intersection Point: This line will intersect the curve at another point, which geometrically represents -(P + 2P) = -3P, due to the symmetric property of elliptic curves.
3. Reflect Across the X-Axis: Reflect this intersection point across the x-axis to find 3P, the result of adding P and 2P.
Continuing the Process to Get 4P, 5P, etc.
- For 4P: Draw a tangent line at 2P (since 2P + 2P = 4P), find the intersection with the curve, and reflect it across the x-axis.
- For 5P: Draw a line through 2P and 3P (since 2P + 3P = 5P), find the intersection with the curve, and reflect it.
This process can be continued indefinitely to calculate multiples of P. Each step involves either drawing a tangent line (for doubling) or a line through two points (for addition), finding the intersection with the curve, and reflecting the point across the x-axis to get the sum.
Geometric vs. Algebraic
Geometrically, this process is elegant and visualizes the group law on elliptic curves. Algebraically, these operations are governed by formulas that calculate the slope of the line (tangent or secant) and use it to find the x-coordinate of the intersection point, and then the y-coordinate. The algebra ensures that the operation is well-defined and consistent across the curve.
This arithmetic underpins many applications of elliptic curves, especially in cryptography, where the difficulty of reversing these operations (finding n given P and nP) forms the basis of elliptic curve cryptography (ECC).