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Removed line numbers in figures

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This commit is contained in:
Aaron Kaiser
2023-06-12 11:53:44 +02:00
parent 9ba0bc2ef3
commit 3df7ccbfe4
10 changed files with 82 additions and 61 deletions
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14
thesis/sections/mu_security_of_eddsa/mu-gamez_implies_mu-uf-nma.tex
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@@ -15,7 +15,7 @@ This section shows that MU-\igame implies MU-UF-NMA security of the EdDSA signat
\hrule
\vspace{1mm}
\large
\begin{algorithmic}[1]
\begin{algorithmic}
\Statex \underline{\game \igame}
\State \textbf{for} $i \in \{1,2,...,N\}$
\State \quad $a_i \randomsample \{2^{n-1}, 2^{n-1} + 2^c, ..., 2^n - 2^c\}$
@@ -24,7 +24,7 @@ This section shows that MU-\igame implies MU-UF-NMA security of the EdDSA signat
\State \Return $\exists (\groupelement{R}^*, \ch^*) \in \pset{Q}, i \in \{1,2,...,N\} \in : \groupelement{R}^* = 2^c s^* \groupelement{B} - 2^c \ch^* \groupelement{A_i}$
\end{algorithmic}
\vspace{2mm}
\begin{algorithmic}[1]
\begin{algorithmic}
\Statex \underline{\oracle \ioracle($\groupelement{R_i} \in \group{G}$)}
\State $\ch_i \randomsample \{0,1\}^{2b}$
\State $\pset{Q} \assign \pset{Q} \cup \{ (\groupelement{R}_i, \ch_i) \}$
@@ -39,7 +39,7 @@ This section shows that MU-\igame implies MU-UF-NMA security of the EdDSA signat
\label{theorem:adv_mu-igame}
Let $\adversary{A}$ be an adversary against MU-\igame. Then,
\[ \advantage{\adversary{A}}{\text{MU-UF-NMA}}(\secparamter) = \advantage{\adversary{B}}{\text{MU-\igame}}(\secparamter) \].
\[ \advantage{\adversary{A}}{\text{MU-UF-NMA}}(\secparamter) = \advantage{\adversary{B}}{\text{MU-\igame}}(\secparamter). \]
\end{theorem}
\paragraph{\underline{Proof Overview}} Like the single-user setting the adversary has to query the random oracle to get the hash value $H(\encoded{R}|\encoded{A_i}|m)$. Again the programmability of the random oracle can be used to embed the challenge from \ioracle oracle into the answer of the random oracle. By embedding the challenge from the \ioracle oracle answer into the answer of the random oracle a valid forgery of the signature also becomes a valid solution for the MU-\igame game.
@@ -50,7 +50,7 @@ This section shows that MU-\igame implies MU-UF-NMA security of the EdDSA signat
\hrule
\begin{multicols}{2}
\large
\begin{algorithmic}[1]
\begin{algorithmic}
\State \underline{\game $G_0$}
\State \textbf{for} $i \in \{1,2,...,N\}$
\State \quad $(h_{i_0}, h_{i_1}, ..., h_{i_{2b-1}}) \randomsample \{0,1\}^{2b}$
@@ -60,7 +60,7 @@ This section shows that MU-\igame implies MU-UF-NMA security of the EdDSA signat
\State \Return $\exists i \in \{1,2,...,N\}: \verify(\groupelement{A_i}, \m^*,\signature^*)$
\end{algorithmic}
\columnbreak
\begin{algorithmic}[1]
\begin{algorithmic}
\Statex \underline{\oracle $H(m \in \{0,1\}^*)$}
\State $\textbf{if } \sum[m] = \bot \textbf{ then}$
\State \quad $\sum[m] \randomsample \{0,1\}^{2b}$
@@ -87,13 +87,13 @@ This section shows that MU-\igame implies MU-UF-NMA security of the EdDSA signat
\hrule
\vspace{1mm}
\large
\begin{algorithmic}[1]
\begin{algorithmic}
\Statex \underline{\textbf{Adversary} $\adversary{B}^{\ioracle(\inp)}(\groupelement{A_1}, \groupelement{A_2}, ..., \groupelement{A_N})$}
\State $(\m^*, \signature^* \assign (\encoded{R}, S)) \randomassign \adversary{A}^{H(\inp)}(\groupelement{A_1}, \groupelement{A_2}, ..., \groupelement{A_N})$
\State \Return $S$
\end{algorithmic}
\vspace{2mm}
\begin{algorithmic}[1]
\begin{algorithmic}
\Statex \underline{\oracle $H(m \in \{0,1\}^*)$}
\State $\textbf{if } \sum[m] = \bot \textbf{ then}$
\State \quad $\textbf{if } \encoded{R} | \encoded{A} | m' \assign m \wedge \groupelement{R}, \groupelement{A} \in \curve \textbf{ then}$