diff --git a/ccs-body.tex b/ccs-body.tex
index 461988d..54e1551 100644
--- a/ccs-body.tex
+++ b/ccs-body.tex
@@ -21,8 +21,7 @@ Recently, Kiayias~\textit{et al.}~\cite{kiayias2021mining} have proposed a block
 However, the security of their protocol was only proven if a fixed PoW difficulty is assumed for ll blocks. This is not a realistic assumption in practice. For example the block difficulty in Bitcoin has shown exponential growth in the past decade. 
 
 
-In this work, we address this important issue and present XX (un petit nom ??), a scheme to construct a succinct representation of the blockchain blockchain using Non-Interactive Proofs-of-Proof-of-Works (NIPoPoWs) that also operates in $O(\polylog(n))$ storage complexity and $O(\polylog(n))$ communication complexity and 
-handles a variable difficulty for the blocks of the blockchain. The main idea of our construction is to XXXXXXXX
+In this work, we address this important issue and present XX (un petit nom ??), a scheme to construct a succinct representation of the blockchain using Non-Interactive Proofs-of-Proof-of-Works (NIPoPoWs) that also operates in $O(\polylog(n))$ storage complexity and $O(\polylog(n))$ communication complexity and handles a variable difficulty for the blocks of the blockchain. The main idea of our construction is to XXXXXXXX
 % In this paper, we focus on the aforementioned protocol.
 % We modify it to fit a variable difficulty setting~\cite{garay2017bitcoin}, with participants joining or leaving the network.
 We prove that the properties needed to maintain security of the protocol still hold in a dynamic context.
@@ -181,7 +180,7 @@ The key idea is to sub-sample the blocks in the blockchain such that the sub-sam
 % \ea{You should explain why a constant and known number of superblocks convinces the verifier}
 %
 The scheme requires every block header to store pointers to the last superblock at every level in order to ensure that the subsampled blocks also form a valid chain. A chain of $n$ blocks will contain superblocks at $O(\log(n))$ levels, as illustrated in Figure~\ref{fig:compression}. Hence, the space and communication complexity of NIPoPoW is $O(\polylog(n))$. 
-The proposal by Kiayias et al.~\cite{10.1145/3460120.3484784}  offers the best-known compression of PoW blockchains so far. It achieves  $O(\polylog(n)c + kd + a)$ storage and communication costs while allowing parties to mine new blocks based on this compressed blockchain, where $k$ is the common prefix parameter, $d$ is the size of application data per block, and $a$ is the size of application data. % in the blockchain. 
+The proposal by Kiayias et al.~\cite{10.1145/3460120.3484784}  offers the best-known compression of PoW blockchains so far. It achieves  $O(\polylog(n)c + kd + a)$ storage and communication costs while allowing parties to mine new blocks based on this compressed blockchain, where $c$ is the size of a block header, $k$ is the common prefix parameter, $d$ is the size of application data per block, and $a$ is the size of application data. % in the blockchain. 
 
 
 \begin{figure}
@@ -232,12 +231,12 @@ Any scheme for operating and compressing blockchains requires to design (i) a \e
 
 \subsubsection{Chain Compression Algorithm}
 
-The Kiayias et al.'s chain compression algorithm (from~\cite{10.1145/3460120.3484784}, Algorithm 1) is parameterized by a security parameter $m$ and the common prefix parameter $k$. System parameter $m$ represents the number of blocks that a party wishes to receive to feel safe. The algorithm compresses the blockchain except for the $k$ most recent blocks, called \emph{unstable} blocks. The compression works as follows: For the highest level $\ell$ that contains more than $2m$ blocks, keep all the blocks but for every level $\mu$ below $\ell$, only keep the last $2m$ blocks and all the blocks after the $m^\text{th}$ block at the $\mu+1$ level. $\Pi$ is used to represent an instance of NIPoPoW proof. %\sg{what is $\mu$ here?} %\ea{We should introduce the $\Pi$ notation here}
+The Kiayias et al.'s chain compression algorithm (from~\cite{10.1145/3460120.3484784}, Algorithm 1) is parameterized by a security parameter $m$ and the common prefix parameter $k$. System parameter $m$ represents the number of blocks that a party wishes to receive to feel safe. The algorithm compresses the blockchain except for the $k$ most recent blocks, called \emph{unstable} blocks. The compression works as follows: for the highest level $\ell$ that contains more than $2m$ blocks, keep all the blocks but for every level $\mu$ below $\ell$, only keep the last $2m$ blocks and all the blocks after the $m^\text{th}$ block at the $\mu+1$ level. $\Pi$ is used to represent an instance of NIPoPoW proof. %\sg{what is $\mu$ here?} %\ea{We should introduce the $\Pi$ notation here}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \subsubsection{Compressed Chain Comparison Algorithm}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %TODO: Give Intuition
-Let $\Pi_1, \Pi_2, \ldots, \Pi_n$ be the different compressed blockchains that a new party receives. The party first applies the compression algorithm to every compressed blockchain to make the comparison fair. To compare any two compressed blockchains $\Pi$ and $\Pi'$, the compression algorithm selects the minimum level $\mu$ that contains a block present in both $\Pi$ and $\Pi'$. If no such block is found, it necessarily implies that the greatest level (compression level $\ell$) in the two compressed blockchains is not the same, and thus simply, the algorithm selects the one with the greatest level. If block $b$ is found in  both $\Pi$ and $\Pi'$ at the same level $\mu$, then the blockchain with the greatest number of blocks after $b$ wins the comparison. 
+Let $\Pi_1, \Pi_2, \ldots, \Pi_n$ be the different compressed blockchains that a new party receives. The party first applies the compression algorithm to every compressed blockchain to make the comparison fair. To compare any two compressed blockchains $\Pi$ and $\Pi'$, the compression algorithm selects the minimum level $\mu$ that contains a block present in both $\Pi$ and $\Pi'$. If no such block is found, it necessarily implies that the greatest level (compression level $\ell$) in the two compressed blockchains is not the same, and thus simply, the algorithm selects the one with the greatest level. If block $b$ is found in both $\Pi$ and $\Pi'$ at the same level $\mu$, then the blockchain with the greatest number of blocks after $b$ wins the comparison. 
 % \section{Mining in Logarithmic Space}
 
 % Prior to presenting our scheme, we briefly describe Kiayias~\textit{et al.}' solution.