reviewed architecture low-level section

Author: EmpieichO

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

@@ -1,19 +1,19 @@
-The Arithmetic State Machine is a secondary state machine that also has an executor (the Arithmetic SM Executor) and an internal Arithmetic program (a set of verification rules written in the PIL language). The Arithmetic SM Executor is available in two languages: Javascript and C/C++.
+The Arithmetic state machine is a secondary state machine that also has an executor (the Arithmetic SM executor) and an internal Arithmetic program (a set of verification rules written in the PIL language). The Arithmetic SM executor is available in two languages: Javascript and C/C++.
 
-It is one of the six secondary state machines receiving instructions from the Main SM Executor. The main purpose of the Arithmetic SM is to carry out elliptic curve arithmetic operations, such as Point Addition and Point Doubling as well as performing 256-bit operations like addition, product or division.
+It is one of the six secondary state machines receiving instructions from the Main SM executor. The main purpose of the Arithmetic SM is to carry out elliptic curve arithmetic operations, such as Point Addition and Point Doubling as well as performing 256-bit operations like addition, product or division.
 
 ## Standard elliptic curve arithmetic
 
 Consider an elliptic curve $E$ defined by $y^2 = x^3 + ax + b$ over the finite field $\mathbb{F} = \mathbb{Z}_p$, where $p$ is the prime,
 
 $$
-p = 2^{256} - 2^{32} - 2^9 - 2^8 - 2^7 -2^6 - 2^4 - 1.
+p = 2^{256} - 2^{32} - 2^9 - 2^8 - 2^7 -2^6 - 2^4 - 1
 $$
 
 Set the coefficients $a = 0$ and $b = 7$, so that $E$ reduces to
 
 $$
-y^2 = x^3 + 7.
+y^2 = x^3 + 7
 $$
 
 ### Point addition
@@ -55,15 +55,15 @@ For instance, if $C = 0$, then $\bf{Eqn\ A}$ states that the result of multiplyi
 
 Or, if $B = 1$,  $\bf{Eqn\ A}$ states that the result of adding $A$ and $C$ is the same as before: $E$ with a carry of $D$. Similarly, division and modular reductions can also be expressed as derivatives of $\bf{Eqn\ A}$.
 
-These operations are performed in the Arithmetic State Machine, with registers satisfying the following PIL relation,
+These operations are performed in the Arithmetic state machine, with registers satisfying the following PIL relation,
 
 $$
 \text{EQ}_0 \colon \quad x_1 \cdot y_1 + x_2 - y_2 \cdot 2^{256} - y_3 = 0
 $$
 
 #### Remark
 
-Since the above Elliptic Curve operations are implemented in the PIL language, it is more convenient to express them in terms of the constraints they must satisfy. These constraints are:
+Since the above elliptic curve operations are implemented in the PIL language, it is more convenient to express them in terms of the constraints they must satisfy. These constraints are:
 
 $$
 \begin{aligned}
@@ -113,22 +113,22 @@ Use the following notation:
 
 $$
 \begin{aligned}
-\mathbf{eq\ } &= x_1[0] \cdot y_1[0] \\
-\mathbf{eq'} &= x_1[0] \cdot y_1[1] + x_1[1] \cdot y_1[0]
+\mathtt{eq\ } &= x_1[0] \cdot y_1[0] \\
+\mathtt{eq'} &= x_1[0] \cdot y_1[1] + x_1[1] \cdot y_1[0]
 \end{aligned}
 $$
 
-But then, the carry generated by $\mathbf{eq}$ has to be taken into account by $\mathbf{eq'}$.
+But then, the carry generated by $\mathtt{eq}$ has to be taken into account by $\mathtt{eq'}$.
 
 Going back to our equations $\text{EQ}_0, \text{EQ}_1, \text{EQ}_2, \text{EQ}_4$; let's see how the operation is performed in $\text{EQ}_0$.
 
-1. Compute $\mathbf{eq}_0 = (x_1[0] \cdot y_1[0]) + x_2[0] - y_3[0]$
+1. Compute $\mathtt{eq}_0 = (x_1[0] \cdot y_1[0]) + x_2[0] - y_3[0]$
 
-2. Compute $\mathbf{eq}_1 = (x_1[0] \cdot y_1[1]) + (x_1[1] \cdot y_1[0]) + x_2[1] - y_3[1]$
+2. Compute $\mathtt{eq}_1 = (x_1[0] \cdot y_1[1]) + (x_1[1] \cdot y_1[0]) + x_2[1] - y_3[1]$
 
-3. $\mathbf{eq}_2 = (x_1[0] \cdot y_1[2]) + (x_1[1] \cdot y_1[1]) + (x_1[2] \cdot y_1[0]) + x_2[2] - y_3[2]$
+3. $\mathtt{eq}_2 = (x_1[0] \cdot y_1[2]) + (x_1[1] \cdot y_1[1]) + (x_1[2] \cdot y_1[0]) + x_2[2] - y_3[2]$
 
-This is continued until one reaches the computation, $\mathbf{eq}_{15} = (x_1[0] \cdot y_1[15]) + (x_1[1] \cdot y_1[14]) + \dots + x_2[15] - y_3[15]$.
+This is continued until one reaches the computation, $\mathtt{eq}_{15} = (x_1[0] \cdot y_1[15]) + (x_1[1] \cdot y_1[14]) + \dots + x_2[15] - y_3[15]$.
 
 At this stage $y_2$ comes into place.
 
@@ -138,9 +138,9 @@ Therefore, we obtain that:
 
 $$
 \begin{aligned}
-\mathbf{eq}_{16} &= (x_1[1]
+\mathtt{eq}_{16} &= (x_1[1]
 \cdot y_1[15]) + (x_1[2] \cdot y_1[14]) + \dots - y_2[0], \\
-\mathbf{eq}_{17} &= (x_1[2] \cdot y_1[15]) + (x_1[3] \cdot y_1[14]) + \dots - y_2[1]
+\mathtt{eq}_{17} &= (x_1[2] \cdot y_1[15]) + (x_1[3] \cdot y_1[14]) + \dots - y_2[1]
 \end{aligned}
 $$
 
@@ -150,15 +150,15 @@ Continuing until the last two:
 
 $$
 \begin{aligned}
-\mathbf{eq}_{30} &= (x_1[15] \cdot y_1[15]) - y_2[14], \\
-\mathbf{eq}_{31} &= -y_2[15].
+\mathtt{eq}_{30} &= (x_1[15] \cdot y_1[15]) - y_2[14], \\
+\mathtt{eq}_{31} &= -y_2[15].
 \end{aligned}
 $$
 
-Now, notice that the carry generated by the $\mathbf{eq}_i$'s is not important for them. That means, if $\mathbf{eq}_i = 10$, then what we really want the result to be is $0$ and save $1$ as a carry for the next operation. To express this fact as a constraint, we say that the following has to be satisfied:
+Now, notice that the carry generated by the $\mathtt{eq}_i$'s is not important for them. That means, if $\mathtt{eq}_i = 10$, then what we really want the result to be is $0$ and save $1$ as a carry for the next operation. To express this fact as a constraint, we say that the following has to be satisfied:
 
 $$
-\mathbf{eq} + \text{carry} = \text{carry}' \cdot 2^{16},
+\mathtt{eq} + \text{carry} = \text{carry}' \cdot 2^{16},
 $$
 
 where $\text{carry}$ represents the carry taken into account in the actual clock, and $\text{carry}'$ represents the carry generated by the actual operation.
@@ -175,6 +175,6 @@ $$
 
 The Polygon zkEVM repository is available on [GitHub](https://github.com/0xPolygonHermez).
 
-Arithmetic SM Executor: [sm_arith folder](https://github.com/0xPolygonHermez/zkevm-proverjs/tree/main/src/sm/sm_arith)
+Arithmetic SM executor: [sm_arith folder](https://github.com/0xPolygonHermez/zkevm-proverjs/tree/main/src/sm/sm_arith)
 
 Arithmetic SM PIL: [arith.pil](https://github.com/0xPolygonHermez/zkevm-proverjs/blob/main/pil/arith.pil)