Minor improvements

thesis/sections/introduction.tex

@@ -12,13 +12,14 @@ 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.
 
+%TODO: Umschreiben
 Tight security proofs are desirable because they provide a better approximation of the bit security of a signature scheme when instantiated with concrete primitives (such as groups or hash functions)\cite{SAC:ChaMenSar11}. A better approximation results in smaller parameters (such as the size of the group) yielding better bit security. This means that, with tighter security proofs, smaller primitives can be used to achieve the same level of security, and smaller primitives often result in more faster computations and therefore more efficient cryptographic schemes.
 
-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 under the discrete logarithm assumption, as shown by Fuchsbauer et al. in \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 under the discrete logarithm assumption, as shown by Fuchsbauer et al. \cite{EC:FucPloSeu20}.
 
 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, introduces 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. While 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.
 
-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} \cite{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 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.
 
 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 under a variant of the one more discrete logarithm assumption, which also takes the key clamping of EdDSA into account.