add first two ZKPs in EC scheme to math.tex

tex-stuff/math.tex

@@ -2,6 +2,37 @@
 \usepackage[a4paper, margin=2cm]{geometry}
 \usepackage{amsmath}
 \begin{document}
+\section{first price auction with tie breaking and private outcome (EC-Version)}
+\subsection{Zero Knowledge Proofs}
+\subsubsection{Proof of Knowledge of a EC DL}
+
+Alice and Bob know $v$ and $g$ with $|g| = n$, but only Alice knows $x$, so that $v = xg$.
+
+\begin{enumerate}
+	\item Alice chooses $z$ at random and calculates $a = zg$.
+	\item Alice computes $c = HASH(g,v,a)$ mod n.
+	\item Alice sends $r = (z + cx)$ mod n and $a$ to Bob.
+	\item Bob checks that $rg = a + cv$.
+\end{enumerate}
+
+\subsection{Proof of equality of tow 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 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}
+
+\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 $m \epsilon \{0, z\}$.
+
 \section{first price auction with tie breaking and private outcome}
 \begin{align}
 	v_{aj} & = \frac{\prod_{i=1}^n \gamma_{aj}^{\times i}}{\prod_{i=1}^n \varphi_{aj}^{\times i}} \\[2.0ex]