Update on Overleaf.

ccs-body.tex

@@ -291,7 +291,7 @@ The problem of blockchain becoming of considerable size was initially predicted
 
 Flyclient~\cite{9152680} allows a succinct and secure construction of proofs in a setting with variable difficulty. They make use of Merkle mountain ranges to reference the whole previous blockchain from every block. If a full node has a proof and mines a new block on top of it, they cannot create a new proof without holding the whole chain. Thus logarithmic space mining is not possible with this scheme.  CoinPrune~\cite{coinprune} still requires to store the entire chain of block header prior to to the pruning point. 
 
-Another approach to built succinct proofs is to rely on SNARKS (for Succinct Non-Interactive Argument of Knowledge). Coda~\cite{coda2020} is such a construction. Coda compresses a chain to polylogarithmic size and updates the proof with new blocks. However, leveraging SNARKs requires a trusted setup for the common reference string. 
+Another approach to build succinct proofs is to rely on SNARKS (for Succinct Non-Interactive Argument of Knowledge). Coda~\cite{coda2020} is such a construction. Coda compresses a chain to polylogarithmic size and updates the proof with new blocks. However, leveraging SNARKs requires a trusted setup for the common reference string. 
 
 Kiayias et al.~\cite{10.1007/978-3-662-53357-4_5} introduced and formalized an interactive proof mechanism, \emph{Proofs-of-Proof-of-Work} (PoPoW) based on superblocks that allows a client to verify a chain in sublinear time and communication complexity. However, the authors later showed the existence of an attack on the scheme and proposed a non-interactive  alternative (NIPoPoWs)~\cite{10.1145/3460120.3484784}. However, the proposed solution did not address the size of the blockchain that needed to be stored by any miner. The authors further used NIPoPoWs to develop a scheme that also allows the miners to operate in $O(\polylog(n))$ storage and communication complexity while reducing the security tolerance to a Byzantine adversary that controls strictly less than a third of the total computation power and limiting itself to operate in an environment with a fixed difficulty~\cite{10.1145/3460120.3484784}.
 
@@ -324,7 +324,7 @@ The main idea of their solution is \emph{(i)} to  attach increasing weights $W_\
 \section{Mining in Logarithmic Space with Variable Difficulty}\label{sec:variable}
 
 \begin{itemize}
-    \item $k$ not correlated to $m$ anymore.
+    \item $k$ is not correlated to $m$ anymore.
     \item Link of $m$ related to $|epoch|$?
 \end{itemize}
 
@@ -534,6 +534,7 @@ Use interactive version for now.
 
 % Theorem 2 Garay variable difficulty
 For the common prefix to hold, we need $k \geq \frac{\theta \gamma m}{4 \tau}$. As $m = 2016$ and $\tau = 4$, we want to compute $\theta$ and $\gamma$ in order to get $k$:
+% We can't right away compute \theta and \gamma with empirical values, as
 \begin{itemize}
     % Definition 5 Garay variable difficulty
     \item $\theta$: we have $f(T_r^{max}(E),n_r) \leq \theta f$ (with $f = 0.03$), so $\theta \geq \frac{f(T_r^{max}(E),n_r)}{f}$. As we can estimate $q$ as $\frac{diff(S)}{rounds(S)}$, with $S$ the number of blocks confirming the weather balloon, $diff(S)$ the expected number of nonces tested to mine the $S$ blocks and $rounds(S)$ the number of rounds occurring during the mining of the $S$ blocks. Then we have $\theta \geq \frac{1 - (1 - \frac{T}{2^\kappa})^{q n_r}}{f}$