Merge pull request 0xPolygon#40 from 0xPolygon/empieichO-docs-review

Update fixed some rendering issues

Author: Nadim Kobeissi

docs/zkEVM/architecture/zkprover/binary-sm.md

@@ -131,10 +131,12 @@ The Plookup checks the operation code, the registries for the input and output 2
 This section provides examples of how the byte-wise operations work.
 
 A $256$-bit integer $\mathbf{a}$ is herein denoted in vector form as $(a_{31}, \dots, a_1, a_0)$ to indicate that,
+
 $$
 \mathbf{a} = a_{31}\cdot (2^8)^{31} + a_{30}\cdot (2^8)^{30} + \cdots + a_1\cdot2^8 + a_0   =
 \sum_{i = {31}}^{0} a_i \cdot (2^8)^i,
 $$
+
 where each $a_i$ is a byte that can take values between $0$ and $2^8 - 1$.
 
 **Example 1**

docs/zkEVM/architecture/zkprover/mem-align-sm.md

@@ -90,7 +90,7 @@ $$
 in the address $\mathtt{0x22}$ of the byte-addressed Ethereum memory. We are using the same $\mathtt{m}_0$ and $\mathtt{m}_1$ (and since we are writing into the same address as before) and they will transition into:
 
 $$
-\mathtt{w}_0 = \mathtt{0x88d1}\color{red}\mathtt{e201e6\dots}\color{black},\quad \mathtt{w}_1 = \mathtt{0x}\color{red} \mathtt{662b}\color{black} \mathtt{ff\dots54f9}.
+\mathtt{w}_0 = \mathtt{0x88d1} \mathtt{e201e6\dots},\quad \mathtt{w}_1 = \mathtt{0x} \mathtt{662b} \mathtt{ff\dots54f9}.
 $$
 
 ![Schema of MSTORE example](../../../img/zkEVM/02mem-align-schm-mstr-eg.png)

docs/zkEVM/architecture/zkprover/memory-sm.md

@@ -14,6 +14,8 @@ Now, let's see the layout in memory of the following two words $\texttt{0xc417..
 
 The below table displays this layout. Let's call this Table 1.
 
+<center>
+
 | $\mathbf{ADDRESS}$ |  $\mathbf{BYTE}$  |
 | :----------------: | :---------------: |
 |    $\mathtt{0}$    |  $\mathtt{0xc4}$  |
@@ -27,6 +29,8 @@ The below table displays this layout. Let's call this Table 1.
 |   $\mathtt{62}$    |  $\mathtt{0xb7}$  |
 |   $\mathtt{63}$    |  $\mathtt{0x23}$  |
 
+</center>
+
 Observe that each word has 32 bytes and the words are stored in Big-Endian form. So, the most significant bytes are set in the lower addresses. The EVM provides three opcodes to interact with the memory area. **There is an opcode to read, and an opcode to write 32-byte words providing an offset**:
 
 - $\texttt{MLOAD}$: It receives an offset and returns the 32 bytes in memory starting at that offset
@@ -36,6 +40,8 @@ Considering our previous memory contents, if we perform an $\texttt{MLOAD}$ with
 
 On the other hand, if we do an $\texttt{MSTORE}$ with an offset of $\texttt{1}$ with the word $\texttt{0x74f0...ce92}$, we would modify the content of the memory as shown in the below table (or Table 2).
 
+<center>
+
 | $\mathbf{ADDRESS}$ |     $\mathbf{BYTE}$      |
 | :----------------: | :----------------------: |
 |    $\mathtt{0}$    |     $\mathtt{0xc4}$      |
@@ -49,6 +55,8 @@ On the other hand, if we do an $\texttt{MSTORE}$ with an offset of $\texttt{1}$
 |   $\mathtt{62}$    |     $\mathtt{0xb7}$      |
 |   $\mathtt{63}$    |     $\mathtt{0x23}$      |
 
+</center>
+
 When the offset is not a multiple of 32 (or 0x20), as in the previous example, we have to use bytes from two different words when doing $\texttt{MLOAD}$ or $\texttt{MSTORE}$.
 
 Finally, the EVM provides a write memory operation that just writes a byte:

docs/zkEVM/architecture/zkprover/paddingkk-bit-sm.md

@@ -170,43 +170,39 @@ Now, the last part of the Padding-KK-Bit SM is to keep track of the last subdivi
 Consider the 256 bits, $\mathtt{0b|10110011\dots1010|\dots|11\dots11|01\dots10101101|}$, where "$|$" separates every set of 32 bits. The table below displays how the $256 + 1$ rows of a $1993$-block are stored in the registers $\{\mathtt{sOut_i}|\ 0 \leq i \leq 7 \}$.
 
 $$
-\small
-\begin{array}{|l|c|c|c|c|c|c|c|c|c|c|c|}
-\hline
-\texttt{sOutBit} & \mathtt{sOut_0} & \mathtt{sOut_1} & \mathtt{\dots} & \mathtt{sOut_7} & \mathtt{FSOut_0} & \mathtt{FSOut_1} & \mathtt{\dots}  & \mathtt{FSOut_7} & \mathtt{latchSOut}\\ \hline
-\text{ }\text{ }\text{ }\text{ } \dots & \dots  & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots\\ \hline
-\quad\text{ }\text{ } 1 & \mathtt{0b1} & \mathtt{0b0} & \dots & \mathtt{0b0} & 1 & 0& \dots & 0 & 0\\
-\quad\text{ }\text{ }0 & \mathtt{0b01} & \mathtt{0b0} & \dots & \mathtt{0b0} & 2 & 0& \dots & 0 & 0 \\
-\quad\text{ }\text{ }1 & \mathtt{0b101} & \mathtt{0b0} & \dots & \mathtt{0b0} & 2^2 & 0&  \dots & 0 & 0\\
-\quad\text{ }\text{ }1 & \mathtt{0b1101} & \mathtt{0b0} & \dots & \mathtt{0b0} & 2^3 & 0& \dots & 0 & 0\\
-\quad\text{ }\text{ }0 & \mathtt{0b01101} & \mathtt{0b0} & \dots & \mathtt{0b0} & 2^4 & 0& \dots & 0 & 0\\
-\quad\text{ }\text{ }1 & \mathtt{0b101101} & \mathtt{0b0} & \dots & \mathtt{0b0} & 2^5 & 0& \dots & 0 & 0\\
-\quad\text{ }\text{ }0 & \mathtt{0b0101101} & \mathtt{0b0} & \dots & \mathtt{0b0} & 2^6 & 0& \dots & 0 & 0\\
-\quad\text{ }\text{ }1 & \mathtt{0b10101101} & \mathtt{0b0} & \dots & \mathtt{0b0} & 2^7 & 0& \dots & 0 & 0\\
-
-\text{ }\text{ }\text{ }\text{ } \dots & \dots & \dots &  \dots & \dots & \dots & \dots & \dots & \dots & \dots\\
-
-\quad\text{ }\text{ }1 & \mathtt{0b1\dots10101101} & \mathtt{0b0} & \dots & \mathtt{0b0} & 2^{30} & 0 & \dots & 0 & 0\\
-\quad\text{ }\text{ }0 & \mathtt{0b01\dots10101101} & \mathtt{0b0} & \dots & \mathtt{0b0} & 2^{31} & 0 & \dots & 0 & 0\\
-\hline
-\quad\text{ }\text{ }1 & \mathtt{0b01\dots10101101} & \mathtt{0b1} & \dots & \mathtt{0b0} & 0 & 1 & \dots & 0 & 0 \\
-\quad\text{ }\text{ }1 & \mathtt{0b01\dots10101101} & \mathtt{0b11} & \dots & \mathtt{0b0} & 0 & 2 & \dots & 0 & 0 \\
-\text{ }\text{ }\text{ }\text{ } \dots & \dots & \dots & \dots& \dots & \dots & \dots & \dots & \dots & \dots \\
-\quad\text{ }\text{ }1 & \mathtt{0b01\dots10101101} & \mathtt{0b1\dots 11} & \dots & \mathtt{0b0} & 0 & 2^{30} & \dots & 0 & 0\\
-\quad\text{ }\text{ }1 & \mathtt{0b01\dots10101101} & \mathtt{0b11\dots 11} & \dots & \mathtt{0b0} & 0 & 2^{31} & \dots & 0 & 0\\
-\hline
-\text{ }\text{ }\text{ }\text{ } \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots\\
-\text{ }\text{ }\text{ }\text{ } \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots\\
-\hline
-\quad\text{ }\text{ }0 & \mathtt{0b01\dots10101101} & \mathtt{0b11\dots 11} & \dots & \mathtt{0b0} & 0 & 0 & \dots & 1 & 0\\
-\quad\text{ }\text{ }1 & \mathtt{0b01\dots10101101} & \mathtt{0b11\dots 11} & \dots & \mathtt{0b10} & 0 & 0 & \dots & 2 & 0\\
-\quad\text{ }\text{ }0 & \mathtt{0b01\dots10101101} & \mathtt{0b11\dots 11} & \dots & \mathtt{0b010} & 0 & 0 &  \dots & 2^2 & 0\\
-\quad\text{ }\text{ }1 & \mathtt{0b01\dots10101101} & \mathtt{0b11\dots 11} & \dots & \mathtt{0b1010} & 0 & 0 &   \dots & 2^3 & 0\\
-\text{ }\text{ }\text{ }\text{ } \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots\\
-\quad\text{ }\text{ }0 & \mathtt{0b01\dots10101101} & \mathtt{0b11\dots 11} & \dots & \mathtt{0b0110011\dots 1010} & 0 & 0 & \dots & 2^{30} & 0\\
-\quad\text{ }\text{ }1 & \mathtt{0b01\dots10101101} & \mathtt{0b11\dots 11} & \dots & \mathtt{0b10110011\dots 1010} & 0 & 0 & \dots & 2^{31} & 1\\ \hline
-
-\end{array}
+    \begin{array}{|l|c|c|c|c|c|c|c|c|c|c|c|}
+    \hline
+    \texttt{sOutBit} & \mathtt{sOut_0} & \mathtt{sOut_1} & \mathtt{\dots} & \mathtt{sOut_7} & \mathtt{FSOut_0} & \mathtt{FSOut_1} & \mathtt{\dots}  & \mathtt{FSOut_7} & \mathtt{latchSOut}\\ \hline
+    \text{ }\text{ }\text{ }\text{ } \dots & \dots  & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots\\ \hline
+    \quad\text{ }\text{ } 1 & \mathtt{0b1} & \mathtt{0b0} & \dots & \mathtt{0b0} & 1 & 0& \dots & 0 & 0\\
+    \quad\text{ }\text{ }0 & \mathtt{0b01} & \mathtt{0b0} & \dots & \mathtt{0b0} & 2 & 0& \dots & 0 & 0 \\
+    \quad\text{ }\text{ }1 & \mathtt{0b101} & \mathtt{0b0} & \dots & \mathtt{0b0} & 2^2 & 0&  \dots & 0 & 0\\
+    \quad\text{ }\text{ }1 & \mathtt{0b1101} & \mathtt{0b0} & \dots & \mathtt{0b0} & 2^3 & 0& \dots & 0 & 0\\
+    \quad\text{ }\text{ }0 & \mathtt{0b01101} & \mathtt{0b0} & \dots & \mathtt{0b0} & 2^4 & 0& \dots & 0 & 0\\
+    \quad\text{ }\text{ }1 & \mathtt{0b101101} & \mathtt{0b0} & \dots & \mathtt{0b0} & 2^5 & 0& \dots & 0 & 0\\
+    \quad\text{ }\text{ }0 & \mathtt{0b0101101} & \mathtt{0b0} & \dots & \mathtt{0b0} & 2^6 & 0& \dots & 0 & 0\\
+    \quad\text{ }\text{ }1 & \mathtt{0b10101101} & \mathtt{0b0} & \dots & \mathtt{0b0} & 2^7 & 0& \dots & 0 & 0\\
+    \text{ }\text{ }\text{ }\text{ } \dots & \dots & \dots &  \dots & \dots & \dots & \dots & \dots & \dots & \dots\\
+    \quad\text{ }\text{ }1 & \mathtt{0b1\dots10101101} & \mathtt{0b0} & \dots & \mathtt{0b0} & 2^{30} & 0 & \dots & 0 & 0\\
+    \quad\text{ }\text{ }0 & \mathtt{0b01\dots10101101} & \mathtt{0b0} & \dots & \mathtt{0b0} & 2^{31} & 0 & \dots & 0 & 0\\
+    \hline
+    \quad\text{ }\text{ }1 & \mathtt{0b01\dots10101101} & \mathtt{0b1} & \dots & \mathtt{0b0} & 0 & 1 & \dots & 0 & 0 \\
+    \quad\text{ }\text{ }1 & \mathtt{0b01\dots10101101} & \mathtt{0b11} & \dots & \mathtt{0b0} & 0 & 2 & \dots & 0 & 0 \\
+    \text{ }\text{ }\text{ }\text{ } \dots & \dots & \dots & \dots& \dots & \dots & \dots & \dots & \dots & \dots \\
+    \quad\text{ }\text{ }1 & \mathtt{0b01\dots10101101} & \mathtt{0b1\dots 11} & \dots & \mathtt{0b0} & 0 & 2^{30} & \dots & 0 & 0\\
+    \quad\text{ }\text{ }1 & \mathtt{0b01\dots10101101} & \mathtt{0b11\dots 11} & \dots & \mathtt{0b0} & 0 & 2^{31} & \dots & 0 & 0\\
+    \hline
+    \text{ }\text{ }\text{ }\text{ } \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots\\
+    \text{ }\text{ }\text{ }\text{ } \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots\\
+    \hline
+    \quad\text{ }\text{ }0 & \mathtt{0b01\dots10101101} & \mathtt{0b11\dots 11} & \dots & \mathtt{0b0} & 0 & 0 & \dots & 1 & 0\\
+    \quad\text{ }\text{ }1 & \mathtt{0b01\dots10101101} & \mathtt{0b11\dots 11} & \dots & \mathtt{0b10} & 0 & 0 & \dots & 2 & 0\\
+    \quad\text{ }\text{ }0 & \mathtt{0b01\dots10101101} & \mathtt{0b11\dots 11} & \dots & \mathtt{0b010} & 0 & 0 &  \dots & 2^2 & 0\\
+    \quad\text{ }\text{ }1 & \mathtt{0b01\dots10101101} & \mathtt{0b11\dots 11} & \dots & \mathtt{0b1010} & 0 & 0 &   \dots & 2^3 & 0\\
+    \text{ }\text{ }\text{ }\text{ } \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots\\
+    \quad\text{ }\text{ }0 & \mathtt{0b01\dots10101101} & \mathtt{0b11\dots 11} & \dots & \mathtt{0b0110011\dots 1010} & 0 & 0 & \dots & 2^{30} & 0\\
+    \quad\text{ }\text{ }1 & \mathtt{0b01\dots10101101} & \mathtt{0b11\dots 11} & \dots & \mathtt{0b10110011\dots 1010} & 0 & 0 & \dots & 2^{31} & 1\\ \hline
+    \end{array}
 $$
 
 Observe how factors $\{\mathtt{FSOut_i} |\ 0 \leq i \leq 7\}$, in the above table, are used to construct the $32$-bit registers $\{\mathtt{sOut_i}|\ 0 \leq \mathtt{i} \leq 7 \}$. Note that the last row contains the complete set of the $256$ bits. These columns fulfil the following relations,