Point doubling and line evaluation

[ColoCarletti]

No description provided.

[ilitteri]

Looks very good! There's a mistake in step 15, the other suggestions are either potential optimizations or variable reuse.

[ilitteri]

If I'm not mistaken, step 4 ($t_{3} = (tmp_{1} + X_{Q})^{2} - tmp_{0} - tmp_{2}$) could be reduced to $t_{3} = U + U$ with $U = tmp_{1}X_{Q}$ using 1 multiplication and 1 addition instead of 3 additions and 1 multiplication.

Dem:
With $tmp_{0} = X_{Q}^{2}$ from step 1, $tmp_{1} = Y_{Q}^{2}$ from step 2, and $tmp_{2} = tmp_{1}^{2} = Y_{Q}^{4}$ from step 3, then
$t_{3} = (tmp_{1} + X_{Q})^{2} - tmp_{0} - tmp_{2} = (Y_{Q}^{2} + X_{Q})^{2} - X_{Q}^{2} - Y_{Q}^{4} = Y_{Q}^{4} + 2Y_{Q}^{2}X_{Q} + X_{Q}^{2} - X_{Q}^{2} - Y_{Q}^{4} = 2Y_{Q}^{2}X_{Q} = Y_{Q}^{2}X_{Q} + Y_{Q}^{2}X_{Q} = tmp_{1}X_{Q} + tmp_{1}X_{Q}$

[ilitteri]

Same here, step 10 could be reduced to $Z_{T} = V + V$ with $V = Y_{Q}Z_{Q}$ using 1 multiplication and 1 addition instead of 3 additions and 2 multiplications.

Dem:
With $tmp_{1} = Y_{Q}^{2}$ from step 2 then
$Z_{T} = (Y_{Q} + Z_{Q})^{2} - tmp_{1} - Z_{Q}^{2} = Y_{Q}^{2} + 2Y_{Q}Z_{Q} + Z_{Q}^{2} - tmp_{1} - Z_{Q}^{2} = Y_{Q}^{2} + 2Y_{Q}Z_{Q} + Z_{Q}^{2} - Y_{Q}^{2} - Z_{Q}^{2} = 2Y_{Q}Z_{Q} = Y_{Q}Z_{Q} + Y_{Q}Z_{Q}$.