add third ZKP formula to math.tex

tex-stuff/math.tex

@@ -15,14 +15,14 @@ Alice and Bob know $v$ and $g$ with $|g| = n$, but only Alice knows $x$, so that
 	\item Bob checks that $rg = a + cv$.
 \end{enumerate}
 
-\subsection{Proof of equality of tow EC DL}
+\subsection{Proof of equality of two EC DL}
 
 Alice and Bob know $v$, $w$, $g_1$ and $g_2$, but only Alice knows $x$, so that
 $v = xg_1$ and $w = xg_2$.
 
 \begin{enumerate}
 	\item Alice chooses $z$ at random and calculates $a = zg_1$ and $b = zg_2$.
-	\item Alice computes $c = HASH(g,v,w,a,b)$ mod n.
+	\item Alice computes $c = HASH(g_1,g_2,v,w,a,b)$ mod n.
 	\item Alice sends $r = (z + cx)$ mod n, $a$ and $b$ to Bob.
 	\item Bob checks that $rg_1 = a + cv$ and $rg_2 = b + cw$.
 \end{enumerate}
@@ -30,8 +30,31 @@ $v = xg_1$ and $w = xg_2$.
 \subsection{Proof that an encrypted value is one out of two values}
 
 Alice proves that an El Gamal encrypted value $(\alpha, \beta) = (m + ry, rg)$
-either decrypts to $0$ or to a fixed value $z$ without revealing which is the
-case, in other words, it is shown that $m \epsilon \{0, z\}$.
+either decrypts to $0$ or to the fixed value $g$ without revealing which is the
+case, in other words, it is shown that $m \epsilon \{0, g\}$.
+
+If $m = 0$:
+
+\begin{enumerate}
+	\item Alice chooses $r_1$, $d_1$, $w$ at random and calculates $a_1 = r_1g + d_1\beta$, $b_1 = r_1y + d_1(\alpha - g)$, $a_2=wg$ and $b_2=wy$.
+	\item Alice computes $c = HASH(g,\alpha,\beta,a_1,b_1,a_2,b_2)$ mod n.
+	\item Alice chooses $d_2=c-d_1$ mod n and $r_2=w-rd_2$ mod n.
+\end{enumerate}
+
+If $m = g$:
+
+\begin{enumerate}
+	\item Alice chooses $r_2$, $d_2$, $w$ at random and calculates $a_1=wg$, $b_1=wy$, $a_2=r_2g + d_2\beta$ and $b_2=r_2y + d_2\alpha$.
+	\item Alice computes $c = HASH(g,\alpha,\beta,a_1,b_1,a_2,b_2)$ mod n.
+	\item Alice chooses $d_1=c-d_2$ mod n and $r_1=w-rd_1$ mod n.
+\end{enumerate}
+
+Then regardless of the value of $m$:
+
+\begin{enumerate}
+	\item Alice sends $(\alpha, \beta), a_1, b_1, a_2, b_2, c, d_1, d_2, r_1, r_2$ to Bob.
+	\item Bob checks that $c=d_1+d_2$ mod n, $a_1=r_1g+d_1\beta$, $b_1=r_1y+d_1(\alpha-g)$, $a_2=r_2g+d_2\beta$ and $b_2=r_2y+d_2\alpha$.
+\end{enumerate}
 
 \section{first price auction with tie breaking and private outcome}
 \begin{align}