Point Doubling

Cryptography · Axiom Academy

Computing P + P = 2P on an elliptic curve using tangent lines 1. Why We Need a Special Formula When adding two distinct points P and Q on an elliptic curve, we draw a line through them. But what happens when P = Q? The two-point formula requires calculating the slope between two different points: When P = Q, both points have the same coordinates (x₁, y₁) = (x₂, y₂), which gives us 0/0 — undefined! 2. Geometric Interpretation: The Tangent Line Instead of a line through two distinct points, we use the tangent line at point P. This tangent line touches the curve at exactly one point (P) and represents the "limiting case" as Q approaches P. The tangent line has the slope equal to the derivative of the curve at point P. This geometric construction gives us the algebraic formulas we need. For an elliptic curve y² = x³ + ax + b, the point doubling formulas are derived using calculus: The slope is the derivative of the curve at P The x-coordinate uses the slope and x₁ The y-coordinate reflects the third intersection point There's an important edge case to consider. What happens when the y-coordinate of P is zero? When y₁ = 0, the tangent line at P is vertical. A vertical line doesn't intersect the curve at any other finite point, so we get: This makes sense geometrically: when P is on the x-axis, the tangent line is vertical and "intersects" the curve at infinity. Let's compute 2P for a concrete example on the curve y² = x³ - 7x + 10 with P = (1, 2).

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