True or false: "Point addition on an elliptic curve is done by simply adding the coordinates — $P(2,4) + Q(3,5) = S(5,9)$."

False — point addition on elliptic curves uses a specific geometric/algebraic formula involving slopes, NOT simple coordinate addition. Adding coordinates produces a point that's generally not even on the curve.

Why coordinate addition is wrong:

• If $P(2,4)$ and $Q(3,5)$ are on curve $y^2 = x^3 + ax + b$, the point $(5,9)$ is almost certainly NOT on the same curve
• EC point addition is defined geometrically: draw a line through P and Q, find the third intersection, reflect

The correct procedure:

1. Compute slope: $\lambda = \frac{y_2 - y_1}{x_2 - x_1} \mod p$
2. $x_3 = \lambda^2 - x_1 - x_2 \mod p$
3. $y_3 = \lambda(x_1 - x_3) - y_1 \mod p$

This is a fundamental misconception. Students must understand that EC "addition" is a completely different operation from regular addition — it's defined by the curve's algebraic structure, not by arithmetic on coordinates.

Go deeper:

• Corbellini — ECC part 2 (the algebraic sum) — why the slope formula, not coordinate addition, defines $P+Q$.
• Elliptic curve point multiplication — Wikipedia — the point-operation formulas.