diff --git a/rfcs/starknet/fri.html b/rfcs/starknet/fri.html
index c74e490..9ba50ad 100644
--- a/rfcs/starknet/fri.html
+++ b/rfcs/starknet/fri.html
@@ -340,24 +340,20 @@ Protocol constants
FRI constants
MAX_LAST_LAYER_LOG_DEGREE_BOUND = 15. The maximum degree of the last layer polynomial (in log2).
MAX_FRI_LAYERS = 15. The maximum number of layers in the FRI protocol.
-MAX_FRI_STEP = 4. The maximum number of layers that can be skipped in FRI (see the overview for more details).
+MAX_FRI_STEP = 4. The maximum number of layers that can be involved in a reduction in FRI (see the overview for more details). This essentially means that each reduction (except for the first as we specify later) can skip 0 to 3 layers.
This means that the standard can be implemented to test that committed polynomials exist and are of degree at most .
-TODO: Step generators
+Step Generators And Inverses
+As explained in the overview, skipped layers must involve the use of elements of the subgroups of order for the number of layers included in a step (from 1 to 4 as specified previously).
+As different generators can generate the same subgroups, we have to define the generators that are expected. Instead, we define the inverse of the generators of groups of different orders (as it more closely matches the code):
-- we are in a coset, so a fixed value
g=3 is chosen
-- we also must understand how to compute the skipped layers, that is what are the elements of the subgroups of order for in used by the prover and what are their corresponding inverses.
+const OMEGA_16: felt252 = 0x5c3ed0c6f6ac6dd647c9ba3e4721c1eb14011ea3d174c52d7981c5b8145aa75;const OMEGA_8: felt252 = 0x446ed3ce295dda2b5ea677394813e6eab8bfbc55397aacac8e6df6f4bc9ca34;const OMEGA_4: felt252 = 0x1dafdc6d65d66b5accedf99bcd607383ad971a9537cdf25d59e99d90becc81e;const OMEGA_2: felt252 = -1
-These are used to skip layers during the FRI protocol. Only 1, 2, 3, or 4 layers can be skipped, each associated to one of the constant below (except for skipping a single layer which is trivial):
-// to skip 4 layers
-const OMEGA_16: felt252 = 0x5c3ed0c6f6ac6dd647c9ba3e4721c1eb14011ea3d174c52d7981c5b8145aa75;
-// to skip 3 layers
-const OMEGA_8: felt252 = 0x446ed3ce295dda2b5ea677394813e6eab8bfbc55397aacac8e6df6f4bc9ca34;
-// to skip 2 layers
-const OMEGA_4: felt252 = 0x1dafdc6d65d66b5accedf99bcd607383ad971a9537cdf25d59e99d90becc81e;
-
-TODO: explain more here
+So here, for example, OMEGA_8 is where is the generator of the subgroup of order that we later use in the Verify A Layer's Query section.
@@ -501,20 +497,20 @@ Verify A Layer's Query
Queries between layers verify that the next layer is computed correctly based on the current layer .
The next layer is either the direct next layer or a layer further away if the configuration allows layers to be skipped.
Specifically, each reduction is allowed to skip 0, 1, 2, or 3 layers (see the MAX_FRI_STEP constant).
-The formula with no skipping is:
+The FRI formula with no skipping is:
- given a layer evaluations at , a query without skipping layers work this way:
- we can compute the next layer's expected evaluation at by computing
- we can then ask the prover to open the next layer's polynomial at that point and verify that it matches
-The formula with 1 layer skipped with the generator of the 4-th roots of unity (such that ):
+The FRI formula with 1 layer skipped with the generator of the 4-th roots of unity (such that ):
As you can see, this requires 4 evaluations of p_{i} at , , , .
-The formula with 2 layers skipped with the generator of the 8-th roots of unity (such that and ):
+The FRI formula with 2 layers skipped with the generator of the 8-th roots of unity (such that and ):
@@ -525,7 +521,7 @@ Verify A Layer's Query
As you can see, this requires 8 evaluations of p_{i} at , , , , , , , .
-The formula with 3 layers skipped with the generator of the 16-th roots of unity (such that , , and ):
+The FRI formula with 3 layers skipped with the generator of the 16-th roots of unity (such that , , and ):
diff --git a/source/starknet/fri.md b/source/starknet/fri.md
index 53909f6..753552e 100644
--- a/source/starknet/fri.md
+++ b/source/starknet/fri.md
@@ -335,27 +335,22 @@ We use the following constants throughout the protocol.
**`MAX_FRI_LAYERS = 15`**. The maximum number of layers in the FRI protocol.
-**`MAX_FRI_STEP = 4`**. The maximum number of layers that can be skipped in FRI (see the overview for more details).
+**`MAX_FRI_STEP = 4`**. The maximum number of layers that can be involved in a reduction in FRI (see the overview for more details). This essentially means that each reduction (except for the first as we specify later) can skip 0 to 3 layers.
This means that the standard can be implemented to test that committed polynomials exist and are of degree at most $2^{15 + 15} = 2^{30}$.
-### TODO: Step generators
+### Step Generators And Inverses
-* we are in a coset, so a fixed value `g=3` is chosen
-* we also must understand how to compute the skipped layers, that is what are the elements of the subgroups of order $2^i$ for $i$ in $[1, 2, 3, 4]$ used by the prover and what are their corresponding inverses.
+As explained in the overview, skipped layers must involve the use of elements of the subgroups of order $2^i$ for $i$ the number of layers included in a step (from 1 to 4 as specified previously).
-These are used to skip layers during the FRI protocol. Only 1, 2, 3, or 4 layers can be skipped, each associated to one of the constant below (except for skipping a single layer which is trivial):
+As different generators can generate the same subgroups, we have to define the generators that are expected. Instead, we define the inverse of the generators of groups of different orders (as it more closely matches the code):
-```rust
-// to skip 4 layers
-const OMEGA_16: felt252 = 0x5c3ed0c6f6ac6dd647c9ba3e4721c1eb14011ea3d174c52d7981c5b8145aa75;
-// to skip 3 layers
-const OMEGA_8: felt252 = 0x446ed3ce295dda2b5ea677394813e6eab8bfbc55397aacac8e6df6f4bc9ca34;
-// to skip 2 layers
-const OMEGA_4: felt252 = 0x1dafdc6d65d66b5accedf99bcd607383ad971a9537cdf25d59e99d90becc81e;
-```
+* `const OMEGA_16: felt252 = 0x5c3ed0c6f6ac6dd647c9ba3e4721c1eb14011ea3d174c52d7981c5b8145aa75;`
+* `const OMEGA_8: felt252 = 0x446ed3ce295dda2b5ea677394813e6eab8bfbc55397aacac8e6df6f4bc9ca34;`
+* `const OMEGA_4: felt252 = 0x1dafdc6d65d66b5accedf99bcd607383ad971a9537cdf25d59e99d90becc81e;`
+* `const OMEGA_2: felt252 = -1`
-TODO: explain more here
+So here, for example, `OMEGA_8` is $1/\omega_8$ where $\omega_8$ is the generator of the subgroup of order $8$ that we later use in the [Verify A Layer's Query](#verify-a-layers-query) section.
## Configuration
@@ -529,13 +524,13 @@ Queries between layers verify that the next layer $p_{i+j}$ is computed correctl
The next layer is either the direct next layer $p_{i+1}$ or a layer further away if the configuration allows layers to be skipped.
Specifically, each reduction is allowed to skip 0, 1, 2, or 3 layers (see the `MAX_FRI_STEP` constant).
-The formula with no skipping is:
+The FRI formula with no skipping is:
* given a layer evaluations at $\pm v$, a query without skipping layers work this way:
* we can compute the next layer's *expected* evaluation at $v^2$ by computing $p_{i+1}(v^2) = \frac{p_{i}(v)+p_{i}(-v)}{2} + \zeta_i \cdot \frac{p_i(v) - p_i(-v)}{2v}$
* we can then ask the prover to open the next layer's polynomial at that point and verify that it matches
-The formula with 1 layer skipped with $\omega_4$ the generator of the 4-th roots of unity (such that $\omega_4^2 = -1$):
+The FRI formula with 1 layer skipped with $\omega_4$ the generator of the 4-th roots of unity (such that $\omega_4^2 = -1$):
* $p_{i+1}(v^2) = \frac{p_{i}(v)+p_{i}(-v)}{2} + \zeta_i \cdot \frac{p_i(v) - p_i(-v)}{2v}$
* $p_{i+1}(-v^2) = \frac{p_{i}(\omega_4 v)+p_{i}(-\omega_4 v)}{2} + \zeta_i \cdot \frac{p_i(v) - p_i(-\omega_4 v)}{2 \cdot \omega_4 \cdot v}$
@@ -543,7 +538,7 @@ The formula with 1 layer skipped with $\omega_4$ the generator of the 4-th roots
As you can see, this requires 4 evaluations of p_{i} at $v$, $-v$, $\omega_4 v$, $-\omega_4 v$.
-The formula with 2 layers skipped with $\omega_8$ the generator of the 8-th roots of unity (such that $\omega_8^2 = \omega_4$ and $\omega_8^4 = -1$):
+The FRI formula with 2 layers skipped with $\omega_8$ the generator of the 8-th roots of unity (such that $\omega_8^2 = \omega_4$ and $\omega_8^4 = -1$):
* $p_{i+1}(v^2) = \frac{p_{i}(v)+p_{i}(-v)}{2} + \zeta_i \cdot \frac{p_i(v) - p_i(-v)}{2v}$
* $p_{i+1}(-v^2) = \frac{p_{i}(\omega_4 v)+p_{i}(-\omega_4 v)}{2} + \zeta_i \cdot \frac{p_i(v) - p_i(-\omega_4 v)}{2 \cdot \omega_4 \cdot v}$
@@ -555,7 +550,7 @@ The formula with 2 layers skipped with $\omega_8$ the generator of the 8-th root
As you can see, this requires 8 evaluations of p_{i} at $v$, $-v$, $\omega_4 v$, $-\omega_4 v$, $\omega_8 v$, $- \omega_8 v$, $\omega_8^3 v$, $- \omega_8^3 v$.
-The formula with 3 layers skipped with $\omega_{16}$ the generator of the 16-th roots of unity (such that $\omega_{16}^2 = \omega_{8}$, $\omega_{16}^4 = \omega_4$, and $\omega_{16}^8 = -1$):
+The FRI formula with 3 layers skipped with $\omega_{16}$ the generator of the 16-th roots of unity (such that $\omega_{16}^2 = \omega_{8}$, $\omega_{16}^4 = \omega_4$, and $\omega_{16}^8 = -1$):
* $p_{i+1}(v^2) = \frac{p_{i}(v)+p_{i}(-v)}{2} + \zeta_i \cdot \frac{p_i(v) - p_i(-v)}{2v}$
* $p_{i+1}(-v^2) = \frac{p_{i}(\omega_4 v)+p_{i}(-\omega_4 v)}{2} + \zeta_i \cdot \frac{p_i(v) - p_i(-\omega_4 v)}{2 \cdot \omega_4 \cdot v}$