Rewritings due to feedback

thesis/sections/introduction.tex

@@ -12,13 +12,13 @@ In a 2020 paper, Brendel et al. showed that Ed25519 satisfies EUF-CMA and SUF-CM
 
 Tightness is a property of a security proof. A security proof is said to be tight if the probability of success of an adversary $\adversary{B}$ attacking problem B, constructed from adversary $\adversary{A}$ attacking problem A, is at most smaller than the probability of success of $\adversary{A}$ by a small constant factor.
 
-Tight security proofs are desirable because they prove the security of multiple instantiations of a cryptographic scheme. In practice, cryptographic schemes are instantiated with primitives that are efficient in order to obtain an overall efficient scheme. If a security proof is not tight, it may not provide meaningful bounds on the security of the scheme, since it may be instantiated with efficient primitives that have parameters too small for the security proof to provide a meaningful bound. The use of less efficient primitives, to which the security proof provides meaningful bounds, may be undesirable for performance reasons.
+Tight security proofs are desirable because they tightly bind the hardness of the underlying assumption to the security of a cryptographic scheme. Without a tight security proof, it is not ruled out that an adversary may be discovered who needs considerably less effort to break the security of a cryptographic scheme compared to adversaries against its underlying assumption \cite{SAC:ChaMenSar11}. For that reason, much larger parameters must be used to securely instantiate the cryptographic scheme compared to the parameters needed to achieve the same level of security in the underlying assumption. This is undesired in practice, as usually a scheme becomes less efficient the larger its parameters are chosen.
 
-For the Schnorr signature scheme, a tight security reduction can be achieved by using the algebraic group model and the random oracle model to directly show the EUF-CMA security using the discrete logarithm assumption, as shown by Fuchsbauer et al. \cite{EC:FucPloSeu20}.
+For the Schnorr signature scheme, a tight security reduction can be achieved by using the algebraic group model and the random oracle model to directly show the EUF-CMA security using the discrete logarithm assumption, as shown by Fuchsbauer et al. \cite{EC:FucPloSeu20}, instead of analyzing it as a canonical identification scheme onto which the Fiat-Schamir transformation is applied.
 
-This is also the approach used in this thesis. A tight security proof for the EdDSA signature scheme can be achieved by utilizing the algebraic group model and random oracle model. However, some details of the EdDSA signature scheme have to be taken into account, which mainly is the different group structure and the key clamping, introduced by the key generation algorithm. Also, the way the signature is parsed has a major impact on the security guarantees of the EdDSA signature scheme. By allowing only one bitstring representation of a scalar, strict parsing ensures SUF-CMA security. Allowing multiple bitstring representations of the same scalar, lax parsing, results only in EUF-CMA security.
+This thesis uses a similar approach to the one in the paper by Fuchsbauer et al. \cite{EC:FucPloSeu20} to achieve a tight security proof for EdDSA. The tight security proof is achieved by utilizing the algebraic group model and the random oracle model. However, some details of the EdDSA signature scheme have to be taken into account, which mainly is the different group structure and the key clamping, introduced by the key generation algorithm. Also, the way the signature is parsed has a major impact on the security guarantees of the EdDSA signature scheme. There are two variations how to parse the signature. One is called strict parsing and the other one is called lax parsing. Strict parsing allows only one bitstring representation of a scalar value, while lax parsing allows multiple bitstring representations of the same scalar value. Strict parsing ensures SUF-CMA security, while lax parsing only ensures EUF-CMA security.
 
-Another important property of a signature scheme, also briefly mentioned in the paper \cite{SP:BCJZ21}, is its multi-security. When looking at practical applications of a signature scheme, not only one user is using the signature scheme, but many users are involved, all of whom have their own key pair. In most cases, an adversary is satisfied with compromising one of the users. This leaves the question of whether an adversary gains an advantage in compromising a single user if he is provided with many public keys and can request signatures for any of the provided public keys. The multi-user security of Schnorr-like signature schemes has been analyzed in several papers \cite{EPRINT:Bernstein15,C:KilMasPan16}, but none of them apply to EdDSA or give a tight reduction.
+Another important property of a signature scheme, also briefly mentioned in \cite{SP:BCJZ21}, is its multi-security. When looking at practical applications of a signature scheme, not only one user is using the signature scheme, but many users are involved, all of whom have their own key pair. In most cases, an adversary is satisfied with compromising one of the users. This leaves the question whether an adversary gains an advantage in compromising a single user if he is provided with many public keys and can request signatures for any of the provided public keys. The multi-user security of Schnorr-like signature schemes has been analyzed in several papers \cite{EPRINT:Bernstein15,C:KilMasPan16}, but none of them apply to EdDSA or give a tight reduction.
 
 This thesis uses the same method of providing a tight security proof in the algebraic group model and the random oracle model to prove the security of EdDSA in the multi-user setting using a variant of the one more discrete logarithm assumption, which also takes the key clamping of EdDSA into account.