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Aaron Kaiser
2023-06-16 12:46:37 +02:00
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thesis/Abschlussarbeit.tex
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@@ -122,7 +122,7 @@ The two main theorems for the single-user security of $\text{EdDSA}_{\text{sp}}$
\[ \advantage{\group{G}, \adversary{A}}{\text{EUF-CMA}}(\secparamter) \leq \advantage{\curve, n, c, L, \adversary{B}}{\sdlog} + \frac{2(\hashqueries + 1)}{2^b} + \frac{\oraclequeries \hashqueries + \oraclequeries}{2^{-\log_2(\lceil \frac{2^{2b} - 1}{L} \rceil 2^{-2b})}} \]
\end{theorem}
The proof begins by showing that the UF-NMA security of EdDSA implies the SUF-CMA/EUF-CMA security of EdDSA with different types of parsing in the random oracle model. With this step, subsequent proofs can be performed without worrying about signature generation, and a unified chain of reduction can be used to prove the security of EdDSA with both parsing variants. Next, an algebraic intermediate game \igame is introduced. This intermediate game serves as a separation for proofs in the random oracle model and those in the algebraic group model. Finally, the intermediate game \igame is reduced to the special discrete logarithm variant \sdlog.
The proof begins by showing that the UF-NMA security of EdDSA implies the SUF-CMA/EUF-CMA security of EdDSA with different types of parsing, in the random oracle model. With this step, subsequent proofs can be performed without worrying about signature generation, and a unified chain of reduction can be used to prove the security of EdDSA with both parsing variants. Next, an algebraic intermediate game \igame is introduced. This intermediate game serves as a separation for proofs in the random oracle model and those in the algebraic group model. Finally, the intermediate game \igame is reduced to the special discrete logarithm variant \sdlog.
The chain of reductions can be depicted as:
@@ -136,11 +136,11 @@ By combining the loss of advantage during all of the proofs above, combined with
\section{The Security of EdDSA in a Multi-User Setting}
In this section the multi-user security of the EdDSA signature scheme will be analyzed. A common approach for Schnorr-like signature schemes is to show it via the Random Self-reducibility property of the canonical identification scheme as done in \cite{C:KilMasPan16}. This approach does not work with the EdDSA signature scheme, since the reduction is not able to rerandomize a public key in a way preserving the distribution of the key generation algorithm. This is due to the fact that valid public key always has to have the n-th bit set.
Now that the single-user security of EdDSA got analyzed, we can take a look at its multi-user security. A common approach for Schnorr-like signature schemes is to show it using the random self-reducibility property of the canonical identification scheme as done in \cite{C:KilMasPan16}. This approach does not work with the EdDSA signature scheme as the underlying identification scheme does not have this random self-reducibility property, since the reduction is not able to rerandomize a public key in a way preserving the distribution of the key generation algorithm. This is due to the fact that valid secret scalar always has to have the n-th bit set.
Therefore, a similar approach to the proof in the single-user setting is used. It is not possible to reduce the \sdlog problem directly since the adversary gets multiple public keys and therefore might not provide a representation of the commitment looking like $\groupelement{R} = r_1 \groupelement{B} + r_2 \groupelement{A}$, which was needed for the discrete logarithm of the public key to be calculated. For this reason a variant of the one-more discrete logarithm assumption (OMDL) has to be used, as introduced in \cite{JC:BNPS03}.
Therefore, a similar approach to the proof in the single-user setting is used. It is not possible to reduce onto the \sdlog problem directly since the adversary gets multiple public keys and therefore might not provide a representation of the commitment looking like $\groupelement{R} = r_1 \groupelement{B} + r_2 \groupelement{A}$, which was needed for the discrete logarithm of the public key to be calculated. For this reason a variant of the one-more discrete logarithm assumption (OMDL) has to be used, as introduced in \cite{JC:BNPS03}.
The proof starts by showing that the MU-UF-NMA security of EdDSA implies MU-SUF-CMA security of EdDSA in the Random Oracle Model. Next an intermediate game is introduced onto which the MU-UF-NMA security of EdDSA is reduced. At last, the security of the intermediate game is reduced onto the security of the variant of the one-more discrete logarithm assumption.
The proof starts by showing that the MU-UF-NMA security of EdDSA implies MU-SUF-CMA security of EdDSA in the random oracle model. Next an intermediate game is introduced onto which the MU-UF-NMA security of EdDSA is reduced. At last, the security of the intermediate game is reduced onto the security of the variant of the one-more discrete logarithm assumption.
The two main theorems for the multi-user security of $\text{EdDSA}_{\text{sp}}$ and $\text{EdDSA}_{\text{lp}}$ are: