[book] Add swap and multiplexer page

book/src/SUMMARY.md

@@ -42,6 +42,7 @@
       - [Optimization for 4/5 bits range check](design/gadgets/decomposition/optimization4_5bits.md)
     - [SHA-256](design/gadgets/sha256.md)
       - [16-bit table chip](design/gadgets/sha256/table16.md)
+    - [Swap and Multiplexer](design/gadgets/swap_multiplexer.md)
 - [Background Material](background.md)
   - [Fields](background/fields.md)
   - [Polynomials](background/polynomials.md)

book/src/design/gadgets/swap_multiplexer.md

@@ -0,0 +1,63 @@
+# Swap and Multiplexer
+
+Swap gates are used to check the Merkle path of a commitment note.
+Multiplixer are used in ZSA Orchard circuit to evaluate note and value commitments
+according to the transaction type (ZSA or native transaction).
+
+## Swap
+
+Given an input pair of field elements $(a,b)$ and a boolean value $swap$,
+we would like to return
+- $(a, b)$ is $swap = 0$, and
+- $(b, a)$ if $swap = 1$.
+
+### Layout
+
+Let $(a_s, b_s) = SWAP(a, b, swap)$.
+We set all values $a_s$, $b_s$, $a$, $b$ and $swap$ on the same row in the layout.
+
+$$
+\begin{array}{|c|c|c|c|c|c|}
+\hline
+a_0 & a_1 & a_2 & a_3 &  a_4 & q_\texttt{swap} \\\hline
+a   & b   & a_s & b_s & swap & 1               \\\hline
+\end{array}
+$$
+
+### Constraints
+
+$$
+\begin{array}{|c|l|}
+\hline
+\text{Degree} & \text{Constraint} \\\hline
+3 & q_\texttt{swap} \cdot \BoolCheck{swap} = 0 \\\hline
+3 & q_\texttt{swap} \cdot (a_s - \Ternary{swap}{b}{a}) = 0 \\\hline
+3 & q_\texttt{swap} \cdot (b_s - \Ternary{swap}{a}{b}) = 0 \\\hline
+\end{array}
+$$
+
+where $\Ternary{swap}{x}{y} = swap \cdot x + (1 - swap) y$.
+
+## Multiplexer
+
+Given two curve points $(x_0, y_0)$ and $(x_1, y_1)$ and a boolean value $choice$,
+we would like to return
+- $(x_0, y_0)$ if $choice=0$, and
+- $(x_1, y_1)$ if $choice=1$.
+
+To perform this $MUX$ operation, we will call twice the $SWAP$ gates once for each coordinate.
+Let $(x, y) = MUX((x_0, y_0), (x_1, y_1), choice)$.
+We have
+$$x=SWAP(x_0, x_1, choice)[0]$$
+$$y=SWAP(y_0, y_1, choice)[0]$$
+
+### Layout
+
+$$
+\begin{array}{|c|c|c|c|c|c|}
+\hline
+a_0 & a_1 & a_2                       & a_3                       &  a_4   & q_\texttt{swap} \\\hline
+x_0 & x_1 & Swap(x_0, x_1, choice)[0] & Swap(x_0, x_1, choice)[1] & choice & 1               \\\hline
+y_0 & y_1 & Swap(y_0, y_1, choice)[0] & Swap(y_0, y_1, choice)[1] & choice & 1               \\\hline
+\end{array}
+$$