...

static/js/modules/interface/proofs.js

@@ -503,7 +503,6 @@ export function verifyFortify(obj, key) {
                 let p = c.verifyNI(verification.zeroProofs[r]);
 
                 if (p !== 0n) {
-                    console.log(p);
                     return false;
                 }
             }

whitepaper/Dissertation.tex

@@ -65,6 +65,8 @@
 %
 %\clearpage
 
+\section{Disambiguation}
+
 \begin{table}[htp]
 	\begin{tabularx}{\textwidth}{c X}
 		\toprule
@@ -72,14 +74,15 @@
 		\\
 		\midrule
 		$|a|$ & Bit length of value $a$ \\
-		$\left(\frac{a}{b}\right)$ & Jacobi symbol for $a, b$ \\
+		$\left(\frac{a}{b}\right)$ & Jacobi symbol for $a, b$ or division (context dependent) \\
-		$\frac{a}{b}$ & Regular division \\
+		$\frac{a}{b}$ & Division \\
 		$\mathbb{Z}_k$ & Additive group of integers modulo $k$ \\
 		$\mathbb{Z}^*_k$ & Multiplicative group of units modulo $k$ \\
 		$\gcd(a, b)$ & Greatest common divisor of $a, b$ \\
 		$\lcm(a, b)$ & Least common multiple of $a, b$ \\
 		$\phi$ & Euler's totient function \\
 		$\lambda$ & Carmichael's totient function \\
+		$H(\dots)$ & Ideal cryptographic hash function \\
 		\bottomrule
 	\end{tabularx}
 \end{table}
@@ -736,7 +739,7 @@ Instead of proving a value is within a range, the prover will demonstrate that a
 		\item Prover transmits $\{ (\psi(R_i), E(n_i, r_i^*)) \mid 0 < i \leq N \}$ where $\psi$ is a random bijection on the regions.
 
 		\item Verifier chooses a random $c \in \{0, 1\}$. \begin{enumerate}
-			\item If $c = 0$, the verifier requests the definition of $\psi$. They then compute the product of the $E(x, r_i) \cdot E(x, r_i^*)$ and verify proofs that each of these is zero.
+			\item If $c = 0$, the verifier requests the definition of $\psi$. They then compute the product of the $E(x, r_i) \cdot E(x, r_i^*)$ and request proofs that each of these is zero.
 
 			\item If $c = 1$, the verifier requests a proof that each $E(n_i, r_i^*)$ is as claimed.
 		\end{enumerate}
@@ -776,7 +779,7 @@ In practice, as we are using Jurik's form of Paillier, the best we can hope for
 
 \subsection{Proving fortifications}
 
-More work is needed on point (5). The range proof alone only works to prevent negative values from appearing in a fortify action. Fortify actions need to be of form $\{ k, -k, 0, \dots, 0 \}$) and the regions corresponding to $k, -k$ amounts must be adjacent.
+Point (5) still remains, as the range proof alone only works to prevent negative values from appearing in a fortify action. Fortify actions need to be of form $\{ k, -k, 0, \dots, 0 \}$) and the regions corresponding to $k, -k$ amounts must be adjacent.
 
 \begin{figure}[htp]
 \centering
@@ -878,12 +881,12 @@ We combine some ideas from the graph isomorphism proofs with ideas from before t
     Run $t$ times in parallel:
 
     \begin{enumerate}
-        \item Prover transmits $\{ (\psi(R_i), E(n_i, r_i^*)) \mid 0 < i \leq N \}$ where $\psi$ is a random bijection on the regions.
+        \item Prover transmits $\{ (\psi(R_i), E(-n_i, r_i^*)) \mid 0 < i \leq N \}$ where $\psi$ is a random bijection on the regions, and $\{ H(R_i, R_j, s_{ij}) \mid R_i \text{ neighbours } R_j \}$ where $s_{ij}$ is a random salt.
 
         \item Verifier chooses a random $c \in \{0, 1\}$. \begin{enumerate}
-            \item If $c = 0$, the verifier requests the definition of $\psi$. They then compute the product of the $E(x, r_i) \cdot E(x, r_i^*)$ and verify proofs that each of these is zero.
+            \item If $c = 0$, the verifier requests the definition of $\psi$ and each salt. They check that the resulting graph is isomorphic to the original graph. They then compute $E(n_i, r_i) \cdot E(-n_i, r_i^*)$ for each $i$ and request a proof that each is zero. Finally, they compute each edge hash and check that there are precisely the correct number of hashes.
 
-            \item If $c = 1$, the verifier requests a proof that each $E(n_i, r_i^*)$ is as claimed.
+            \item If $c = 1$, the verifier requests proofs that $|S| - 2$ are zero and that the remaining pair add to zero. They then request the salt used to produce the hash along the edge joining the two non-zero elements, and test that this hash is correct.
         \end{enumerate}
     \end{enumerate}
 \end{protocol}