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used uniform font for sets

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This commit is contained in:
Aaron Kaiser
2023-05-15 10:12:44 +02:00
parent 66ab947000
commit 048dd04b86
7 changed files with 31 additions and 31 deletions
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14
thesis/sections/security_of_eddsa/uf-nma_implies_suf-cma.tex
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@@ -47,7 +47,7 @@ The proof starts by providing an algorithm which generates correctly distributed
\State $s \leftarrow 2^n + \sum_{i=c}^{n-1} 2^i h_i$
\State $\groupelement{A} \assign s \groupelement{B}$
\State $(\m^*, \signature^*) \randomassign \adversary{A}^{H(\inp), \sign(\inp)}(\groupelement{A})$
\State \Return $\verify(\groupelement{A}, \m^*,\signature^*) \wedge (\m^*, \signature^*) \notin Q$
\State \Return $\verify(\groupelement{A}, \m^*,\signature^*) \wedge (\m^*, \signature^*) \notin \pset{Q}$
\end{algorithmic}
\columnbreak
\begin{algorithmic}[1]
@@ -73,7 +73,7 @@ The proof starts by providing an algorithm which generates correctly distributed
\State $S \assign (r + s\sum[\encoded{R} | \encoded{A} | \m]) \pmod L$
\EndBox
\State $\signature \assign (\encoded{R}, S)$
\State $Q \assign Q \cup \{(\m, \signature)\}$
\State $\pset{Q} \assign \pset{Q} \cup \{(\m, \signature)\}$
\State \Return $\signature$
\end{algorithmic}
\end{multicols}
@@ -96,7 +96,7 @@ The proof starts by providing an algorithm which generates correctly distributed
\State \quad $abort$
\State $\sum[\encoded{R} | \encoded{A} | \m] = \textbf{ch}$
\State $\signature \assign (\encoded{R}, S)$
\State $Q \assign Q \cup \{(\m, \signature)\}$
\State $\pset{Q} \assign \pset{Q} \cup \{(\m, \signature)\}$
\State \Return $\signature$
\EndBox
\end{algorithmic}
@@ -123,7 +123,7 @@ The proof starts by providing an algorithm which generates correctly distributed
\[ \Pr[G_2^{\adversary{A}} \Rightarrow 1] = \Pr[G_3^{\adversary{A}} \Rightarrow 1]. \]
\item Finally, Game $G_3$ is well prepared to show that there exists an adversary $\adversary{B}$ satisfying
\item Finally, Game $G_3$ is well-prepared to show that there exists an adversary $\adversary{B}$ satisfying
\begin{align}
\Pr[G_3^{\adversary{A}} \Rightarrow 1] = \advantage{\adversary{B}}{\text{UF-NMA}}(\secparamter). \label{eq:adv_uf-nma}
@@ -147,7 +147,7 @@ The proof starts by providing an algorithm which generates correctly distributed
\State \quad $abort$
\State $\sum[\encoded{R} | \encoded{A} | m] = \textbf{ch}$
\State $\signature \assign (\encoded{R}, S)$
\State $Q \assign Q \cup \{(\m, \signature)\}$
\State $\pset{Q} \assign \pset{Q} \cup \{(\m, \signature)\}$
\State \Return $\signature$
\end{algorithmic}
\end{multicols}
@@ -178,7 +178,7 @@ The proof starts by providing an algorithm which generates correctly distributed
\subsection{UF-NMA $\Rightarrow$ $\text{EUF-CMA}_{\text{EdDSA with lax parsing}}$ (ROM)}
This section shows that the UF-NMA security of EdDSA implies the EUF-CMA security of EdDSA with lax parsing using the random oracle model. This proof is very similar to the proof of the SUF-CMA security of EdDSA with strict parsing. The modification of the games are the same as in the proof above with the only difference being the win condition, which is $\verify(\groupelement{A}, \m^*,\signature^*) \wedge \m^* \notin Q$. For this reason this proofs starts at showing the existence of an adversary $\adversary{B}$ breaking UF-NMA security.
This section shows that the UF-NMA security of EdDSA implies the EUF-CMA security of EdDSA with lax parsing using the random oracle model. This proof is very similar to the proof of the SUF-CMA security of EdDSA with strict parsing. The modification of the games are the same as in the proof above with the only difference being the win condition, which is $\verify(\groupelement{A}, \m^*,\signature^*) \wedge \m^* \notin \pset{Q}$. For this reason this proofs starts at showing the existence of an adversary $\adversary{B}$ breaking UF-NMA security.
\begin{theorem}
\label{theorem:adv2_uf-nma}
@@ -213,7 +213,7 @@ This section shows that the UF-NMA security of EdDSA implies the EUF-CMA securit
\State \quad $abort$
\State $\sum[\encoded{R} | \encoded{A} | m] = \textbf{ch}$
\State $\signature \assign (\encoded{R}, S)$
\State $Q \assign Q \cup \{\m\}$
\State $\pset{Q} \assign \pset{Q} \cup \{\m\}$
\State \Return $\signature$
\end{algorithmic}
\end{multicols}