Walk through a concrete point addition on $y^2 \equiv x^3 + 8x + 5 \mod 11$ with $P(0, 4)$ and $Q(3, 1)$.

Compute slope $\lambda = \frac{1-4}{3-0} = \frac{-3}{3} = (-3) \cdot 3^{-1} \equiv (-3) \cdot 4 \equiv 10 \mod 11$. Then $x_3 = 10^2 - 0 - 3 = 97 \equiv 9 \mod 11$ and $y_3 = 10(0 - 9) - 4 = -94 \equiv 5 \mod 11$. Result: $R = (9, 5)$.

Given: Curve $y^2 = x^3 + 8x + 5 \mod 11$, $P(0, 4)$, $Q(3, 1)$

Step 1 — Slope: $$\lambda = \frac{y_2 - y_1}{x_2 - x_1} \mod 11 = \frac{1 - 4}{3 - 0} \mod 11 = \frac{-3}{3} \mod 11$$

Need $3^{-1} \mod 11$: since $3 \cdot 4 = 12 \equiv 1 \mod 11$, we get $3^{-1} = 4$.

$$\lambda = (-3) \cdot 4 \mod 11 = -12 \mod 11 = 10$$

Step 2 — x-coordinate: $$x_3 = \lambda^2 - x_1 - x_2 \mod 11 = 100 - 0 - 3 = 97 \mod 11 = 9$$

Step 3 — y-coordinate: $$y_3 = \lambda(x_1 - x_3) - y_1 \mod 11 = 10 \cdot (0 - 9) - 4 = -94 \mod 11 = 5$$

Result: $P + Q = (9, 5)$

Verification: Check $(9, 5)$ is on the curve: $5^2 = 25 \equiv 3 \mod 11$. $9^3 + 8 \cdot 9 + 5 = 729 + 72 + 5 = 806 \equiv 3 \mod 11$. ✓

Common mistakes to avoid:

• Never simplify/cancel fractions in modular arithmetic — always compute the modular inverse
• Watch negative numbers: $-94 \mod 11 = -94 + 9 \cdot 11 = 5$
• The notation $\frac{a}{b} \mod p$ means $a \cdot b^{-1} \mod p$, not regular division

Go deeper:

• Corbellini — ECC part 2 (worked point addition over $\mathbb{F}_p$) — the same secant-slope computation, step by step.