# Item List

## Associativity 2

Associativity is a fundamental property for the arithmetic of points on an elliptic curve, ensuring that for any three points P, Q, and R on the curve, the equation (P + Q) + R = P + (Q + R) always holds.

## Associativity 1

Adding two points on an elliptic curve involves a specific process defined by the curve's geometry and algebraic properties.

## 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.

## Doubling Points - 2P, 4P, 8P, 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 tangent line to that point, which then crosses the curve at the point S. We take -S = 2R, and find the tangent line to that point, which then crosses the curve at the point T. etc. This effectively "doubles" P each time.

## The Neutral Element

The neutral element of an elliptic curve, often referred to in the context of the curve's group structure, is a fundamental concept that plays a pivotal role in the arithmetic defined on the curve.