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more proofs

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This commit is contained in:
Aaron Kaiser
2023-04-20 12:03:33 +02:00
parent 0baf01b6ca
commit f527b43068
6 changed files with 166 additions and 22 deletions
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18
thesis/sections/mu_security_of_eddsa/mu-gamez_implies_mu-uf-nma.tex
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@@ -6,7 +6,7 @@ This section shows that MU-\igame implies MU-UF-NMA security of the EdDSA signat
%TODO: Fix collision
\begin{definition}[MU-\igame]
Let $n$ and $n$ be positive integers. For an adversary $\adversary{A}$ we define its advantage in the MU-\igame as following:
Let $n$ and $N$ be positive integers. For an adversary $\adversary{A}$ we define its advantage in the MU-\igame as following:
\[ \advantage{\adversary{A}}{\text{MU-\igame}}(\secparamter) \assign | \Pr[\text{MU-\igame}^{\adversary{A}} \Rightarrow 1] | \].
\end{definition}
@@ -17,11 +17,11 @@ This section shows that MU-\igame implies MU-UF-NMA security of the EdDSA signat
\large
\begin{algorithmic}[1]
\Statex \underline{\game \igame}
\State \textbf{for} $i \in \{1,2,...,n\}$
\State \textbf{for} $i \in \{1,2,...,N\}$
\State \quad $a_i \randomsample \{2^{n-1}, 2^{n-1} + 8, ..., 2^n - 8\}$
\State \quad $\groupelement{A_i} \assign a_i \groupelement{B}$
\State $s^* \randomsample \adversary{A}^{\ioracle(\inp)}(\groupelement{A_1}, \groupelement{A_2}, ..., \groupelement{A_n})$
\State \Return $\exists (\groupelement{R}^*, \ch^*) \in Q, i \in \{1,2,...,n\} \in : \groupelement{R}^* = 2^c s^* \groupelement{B} - 2^c \ch^* \groupelement{A_i}$
\State $s^* \randomsample \adversary{A}^{\ioracle(\inp)}(\groupelement{A_1}, \groupelement{A_2}, ..., \groupelement{A_N})$
\State \Return $\exists (\groupelement{R}^*, \ch^*) \in Q, i \in \{1,2,...,N\} \in : \groupelement{R}^* = 2^c s^* \groupelement{B} - 2^c \ch^* \groupelement{A_i}$
\end{algorithmic}
\vspace{2mm}
\begin{algorithmic}[1]
@@ -52,12 +52,12 @@ This section shows that MU-\igame implies MU-UF-NMA security of the EdDSA signat
\large
\begin{algorithmic}[1]
\State \underline{\game $G_0$}
\State \textbf{for} $i \in \{1,2,...,n\}$
\State \textbf{for} $i \in \{1,2,...,N\}$
\State \quad $(h_{i_0}, h_{i_1}, ..., h_{i_{2b-1}}) \randomsample \{0,1\}^{2b}$
\State \quad $s_i \leftarrow 2^n + \sum_{i=c}^{n-1} 2^i h_i$
\State \quad $\groupelement{A_i} \assign s_i \groupelement{B}$
\State $(\m^*, \signature^*) \randomassign \adversary{A}^{H(\inp)}(\groupelement{A_1}, \groupelement{A_2}, ..., \groupelement{A_n})$
\State \Return $\exists i \in \{1,2,...,n\}: \verify(\groupelement{A_i}, \m^*,\signature^*)$
\State $(\m^*, \signature^*) \randomassign \adversary{A}^{H(\inp)}(\groupelement{A_1}, \groupelement{A_2}, ..., \groupelement{A_N})$
\State \Return $\exists i \in \{1,2,...,N\}: \verify(\groupelement{A_i}, \m^*,\signature^*)$
\end{algorithmic}
\columnbreak
\begin{algorithmic}[1]
@@ -88,8 +88,8 @@ This section shows that MU-\igame implies MU-UF-NMA security of the EdDSA signat
\vspace{1mm}
\large
\begin{algorithmic}[1]
\Statex \underline{\textbf{Adversary} $\adversary{B}^{\ioracle(\inp)}(\groupelement{A_1}, \groupelement{A_2}, ..., \groupelement{A_n})$}
\State $(\m^*, \signature^* \assign (\encoded{R}, S)) \randomassign \adversary{A}^{H(\inp)}(\groupelement{A_1}, \groupelement{A_2}, ..., \groupelement{A_n})$
\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}