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diff --git a/tex-stuff/math.tex b/tex-stuff/math.tex index 4a990cb..09046d2 100644 --- a/tex-stuff/math.tex +++ b/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] |