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book: Remove superfluous checkmarks 

Co-authored-by: Daira Hopwood <daira@jacaranda.org>
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str4d 2021-07-01 13:46:48 +01:00 committed by GitHub
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14
book/src/design/circuit/gadgets/ecc/addition.md
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@ -17,12 +17,12 @@ Substituting for $\lambda$, we get the constraints:
## Complete addition
$\begin{array}{rcll}
\mathcal{O} &+& \mathcal{O} &= \mathcal{O} ✓\\
\mathcal{O} &+& (x_q, y_q) &= (x_q, y_q) ✓\\
(x_p, y_p) &+& \mathcal{O} &= (x_p, y_p) ✓\\
(x, y) &+& (x, y) &= [2] (x, y) ✓\\
(x, y) &+& (x, -y) &= \mathcal{O} ✓\\
(x_p, y_p) &+& (x_q, y_q) &= (x_p, y_p) \;⸭\; (x_q, y_q), \text{if } x_p \neq x_q ✓
\mathcal{O} &+& \mathcal{O} &= \mathcal{O} \\
\mathcal{O} &+& (x_q, y_q) &= (x_q, y_q) \\
(x_p, y_p) &+& \mathcal{O} &= (x_p, y_p) \\
(x, y) &+& (x, y) &= [2] (x, y) \\
(x, y) &+& (x, -y) &= \mathcal{O} \\
(x_p, y_p) &+& (x_q, y_q) &= (x_p, y_p) \;⸭\; (x_q, y_q), \text{if } x_p \neq x_q
\end{array}$
Suppose that we represent $\mathcal{O}$ as $(0, 0)$. ($0$ is not an $x$-coordinate of a valid point because we would need $y^2 = x^3 + 5$, and $5$ is not square in $\mathbb{F}_q$. Also $0$ is not a $y$-coordinate of a valid point because $-5$ is not a cube in $\mathbb{F}_q$.)
@ -334,4 +334,4 @@ $$
\delta &= 0 \\
\end{aligned}
$$
because in all remaining cases, $x_q \neq x_p, x_p \neq 0, x_q \neq 0.$
because in all remaining cases, $x_q \neq x_p, x_p \neq 0, x_q \neq 0.$