diff --git a/Paper.tex b/Paper.tex
index b910398f..aceee2a4 100644
--- a/Paper.tex
+++ b/Paper.tex
@@ -1481,6 +1481,62 @@ \subsection{zkSNARK Related Precompiled Contracts}
 \end{eqnarray}
 We define $G_2$ to be the subgroup of $(C_2,+)$ generated by $P_2$. $G_2$ is known to be a cyclic group of order $q$. For a point $P$ in $G_2$, we define $\log_{P_2}(P)$ be the smallest natural number $n$ satisfying $n\cdot P_2=P$. With this definition, $\log_{P_2}(P)$ is at most $q-1$.
 
+A 32 byte number $\mathbf{x}\in\mathbf{P}_{256}$ might and might not represent an element of $F_p$.
+\begin{equation}
+\delta_p(\mathbf x)\equiv\begin{cases}
+\mathbf x&\text{if}\ \mathbf x<p\\
+\varnothing&\text{otherwise}
+\end{cases}
+\end{equation}
+
+A 64 byte data $\mathbf x\in\mathbf B_{512}$ might and might not represent an element of $G_1$.
+\begin{eqnarray}
+\delta_1(\mathbf x)&\equiv&\begin{cases}
+g_1&\text{if}\ g_1\in G_1\\
+\varnothing&\text{otherwise}
+\end{cases}\\
+g_1&\equiv&\begin{cases}
+(x,y)&\text{if}\ x\neq\varnothing\wedge y\neq\varnothing\\
+\varnothing&\text{otherwise}
+\end{cases}\\
+x&\equiv&\delta_p(\mathbf x[0..31])\\
+y&\equiv&\delta_p(\mathbf x[32..63])
+\end{eqnarray}
+
+A 128 byte data $\mathbf x\in\mathbf B_{1024}$ might and might not represent an element of $G_2$.
+\begin{eqnarray}
+\delta_2(\mathbf x)&\equiv&\begin{cases}
+g_2&\text{if}\ g_2\in G_2\\
+\varnothing&\text{otherwise}
+\end{cases}\\
+g_2&\equiv&\begin{cases}
+((x_0i+y_0),(x_1i+y_1))&\text{if}\ x_0\neq\varnothing\wedge y_0\neq\varnothing\wedge x_1\neq\varnothing\wedge y_1\neq\varnothing\\
+\varnothing&\text{otherwise}
+\end{cases}\\
+x_0&\equiv&\delta_p(\mathbf x[0..31])\\
+y_0&\equiv&\delta_p(\mathbf x[32..63])\\
+x_1&\equiv&\delta_p(\mathbf x[64..95])\\
+y_1&\equiv&\delta_p(\mathbf x[96..127])
+\end{eqnarray}
+
+We define a precompiled contract for zkSNARK verification.
+\begin{eqnarray}
+\Xi_{\mathtt{SNARKV}}&\equiv&\Xi_{\mathtt{PRE}}\quad\text{except:}\\
+\qquad\Xi_{\mathtt{SNARKV}}(\boldsymbol\sigma,g,I)&=&(\varnothing,0,A^0,())\quad\text{if}\ F\\
+F&\equiv&(|I_\mathbf{d}|\bmod 192\neq 0\vee(\exists j.\ a_j=\varnothing\vee b_j=\varnothing))\\
+g_r&=&?\\
+\mathbf o[0..31]&\equiv&\begin{cases}
+0x0000000000000000000000000000000000000000000000000000000000000001&\text{if}\ v\wedge\neg F\\
+0x0000000000000000000000000000000000000000000000000000000000000000&\text{if}\ \neg v\wedge\neg F
+\end{cases}\\
+v&\equiv&(\log_{P_1}(a_1)\times\log_{P_2}(b_1)+\cdots+\log_{P_1}(a_k)\times\log_{P_2}(b_k)\equiv 0\mod q)\\
+a_1&\equiv&\delta_1(I_{\mathbf d}[0..63])\\
+b_1&\equiv&\delta_2(I_{\mathbf d}[64..191])\\\nonumber
+\vdots\\
+a_k&\equiv&\delta_1(I_{\mathbf d}[(|I_{\mathbf d}|-192)..(|I_{\mathbf d}|-129)])\\
+b_k&\equiv&\delta_2(I_{\mathbf d}[(|I_{\mathbf d}|-128)..(|I_{\mathbf d}|-1)])
+\end{eqnarray}
+
 \section{Signing Transactions}\label{app:signing}
 
 The method of signing transactions is similar to the `Electrum style signatures'; it utilises the SECP-256k1 curve as described by \cite{gura2004comparing}.