O(log n) changed to \mathcal{O}(log n)

Author: Maltesius

report/src/sections/04-Approach.tex

@@ -399,7 +399,7 @@ \subsubsection{Size reduction}
 We first set our focus on Curdleproofs, as this is the protocol we have modified directly.
 As mentioned in~\autoref{sec:background-zkps}, the size of Curdleproofs is $18+10 \log(\ell+4)\mathbb{G}$, $7\mathbb{F}$.
 The dependence on the $\log$ stems from the number of recursive rounds that take place in the~\gls{sameperm} and~\gls{samemsm} proofs.
-In the proof of theorem 1, we show CAAUrdleproofs to be $O(\log n)$.
+In the proof of theorem 1, we show CAAUrdleproofs to be $\mathcal{O}(\log n)$.
 This means that the size of CAAUrdleproofs must be $18+10 \lceil\log(\ell+4)\rceil\mathbb{G}$, $7\mathbb{F}$.
 
 CAAUrdleproofs therefore has the same proof size as Curdleproofs.

report/src/sections/appendix/02-thm1proof.tex

@@ -32,7 +32,7 @@ \section{Proof of Theorem 1}\label{sec:appendix-thm1proof}
     \paragraph*{\textbf{Proof of knowledge-soundness and completeness}}
     For soundness and completeness, we refer to Theorem 3 of Springproofs~\cite{zhang2024springproofs}.
     \begin{theorem}[Springproofs Theorem 3]
-        Given a terminative SIPA$(f)$, if the number of compression steps in SIPA$(f)$ is $O(\log n)$, then SIPA$(f)$ is a complete and computational knowledge sound argument of relation (1).
+        Given a terminative SIPA$(f)$, if the number of compression steps in SIPA$(f)$ is $\mathcal{O}(\log n)$, then SIPA$(f)$ is a complete and computational knowledge sound argument of relation (1).
         Moreover, the Fiat-Shamir transformation of SIPA$(f)$ is a non-interactive random oracle argument having completeness and computational knowledge soundness as well.
     \end{theorem}
     Here, relation (1) is
@@ -84,7 +84,7 @@ \section{Proof of Theorem 1}\label{sec:appendix-thm1proof}
     \end{align}
     This exactly the same commitment as in~\autoref{al:P}.
 
-    Therefore, using Curdleproofs' DL~\gls{ipa} and the pre-compression scheme function, we can instantiate SIPA$(f)$, equivalent to CAAUrdleproofs, as a terminative SIPA$(f)$, with $O(\log n)$ compression steps.
+    Therefore, using Curdleproofs' DL~\gls{ipa} and the pre-compression scheme function, we can instantiate SIPA$(f)$, equivalent to CAAUrdleproofs, as a terminative SIPA$(f)$, with $\mathcal{O}(\log n)$ compression steps.
     Hence, SIPA$(f)$ is a complete and computational knowledge sound argument of relation (1).
     We have just shown that Curdleproofs'~\gls{ipa} proves the same relation, so the properties hold for our SIPA$(f)$ as well.
     Furthermore, Curdleproofs uses the Fiat-Shamir transformation for its verifier challenges.