fix host functions specidifcations

neps/nep-0488.md

@@ -503,7 +503,7 @@ The function calculates the sum of signed elements on the BLS12-381 curve. It ac
 
 The operations, including the $E(F_p)$ curve, points on the curve, multiplication by -1, and the addition operation, are detailed in the BLS12-381 Curve Specification section.
 
-Note: The function accepts points from the entire curve, not restricted to $G_1$.
+Note: This function accepts points from the entire curve and is not restricted to points in $G_1$.
 
 ***Input:***
 
@@ -523,7 +523,7 @@ The ERROR_CODE is returned.
 
 <ins>Tests for the sum of two points</ins>
 
-This section aims to verify the correctness of summing two valid elements on the curve:
+This section aims to verify the correctness of summing up two valid elements on the curve:
 
 - Utilize points on the curve with known addition results for comparison, such as tests from EIP-2537[^47],[^48].
 - Generate random points on the curve and verify the commutative property: P + Q = Q + P.
@@ -564,16 +564,16 @@ This section aims to validate the handling of incorrect input data:
 - Erroneous coding of field elements resulting in a correct element on the curve modulo p.
 - Erroneous coding of field elements with an incorrect extra bit in the decompressed encoding.
 - Point not on the curve.
-- Incorrect encoding of the point on infinity.
+- Incorrect encoding of the point at infinity.
 - Input is beyond memory bounds.
 
 <ins>Tests for the sum of an arbitrary amount of points</ins>
 
 This section focuses on validating the summation functionality with an arbitrary number of points:
 
-- Generate random points on the curve and verify that the sum of the random permutation matches.
+- Generate random points on the curve and verify that the sum of a random permutation matches.
 - Generate random points on the curve and utilize another library to validate results.
-- Create points and cross-check the outcome with the multiexp function.
+- Create points and cross-check the outcome with the `multiexp` function.
 - Generate random points from $G_1$ and confirm that the sum is also from $G_1$.
 
 Edge cases:
@@ -599,7 +599,7 @@ The function computes the sum of the signed elements on the BLS12-381 curve. It
 
 The $E'(F_{p^2})$ curve, the points on the curve, the multiplication by -1, and the addition operation are all defined in the BLS12-381 Curve Specification section.
 
-Note: The function accepts any points on the curve, not limited to $G_2$.
+Note: The function accepts any points on the curve and is not limited to points in $G_2$.
 
 ***Input:*** 
 
@@ -613,7 +613,7 @@ More details are available in the Curve Points Encoding section.
 The ERROR_CODE is returned.
 
 - ERROR_CODE = 0: the input is correct
-  - <ins>Output:</ins> 192 bytes represent one point $\in E'(F_{p^2})$ in its decompressed form. In case of an empty input, it outputs a point on infinity (refer to the Curve Points Encoding section for more details).
+  - <ins>Output:</ins> 192 bytes represent one point $\in E'(F_{p^2})$ in its decompressed form. In case of an empty input, it outputs the point at infinity (refer to the Curve Points Encoding section for more details).
 - ERROR_CODE = 1:
   - Points or signs are incorrectly encoded (refer to Curve points encoded section).
   - Point is not on the curve.
@@ -658,7 +658,7 @@ Please note:
 The ERROR_CODE is returned.
 
 - ERROR_CODE = 0: the input is correct
-  - <ins>Output:</ins> 96 bytes represent one point $\in E(F_p)$ in its decompressed form. In case of an empty input, it outputs a point on infinity (refer to the Curve Points Encoding section for more details).
+  - <ins>Output:</ins> 96 bytes represent one point $\in E(F_p)$ in its decompressed form. In case of an empty input, it outputs the point at infinity (refer to the Curve Points Encoding section for more details).
 - ERROR_CODE = 1:
   - Points are incorrectly encoded (refer to Curve points encoded section).
   - Point is not on the curve.
@@ -741,14 +741,14 @@ Please note:
 
 ***Input:*** the sequence of pairs $(p_i, s_i)$, where $p_i \in E'(F_{p^2})$ is a point on the curve and $s_i \in \mathbb{N}_0$ is a scalar. 
 
-Expected `224*k` bytes as an input that is interpreted as byte concatenation of `k` slices, each slice is the concatenation of uncompressed point from $E'(F_{p^2})$ — `192` bytes and scalar — `32` bytes. More details are in the Curve Points Encoding section.
+The expected input size is `224*k` bytes, interpreted as the byte concatenation of `k` slices. Each slice is the concatenation of an uncompressed point from $E'(F_{p^2})$ — `192` bytes and a scalar — `32` bytes. More details are in the Curve Points Encoding section.
 
 ***Output:*** 
 
 The ERROR_CODE is returned.
 
 - ERROR_CODE = 0: the input is correct
-  - <ins>Output:</ins> 192 bytes represent one point $\in E'(F_{p^2})$ in its decompressed form. In case of an empty input, it outputs a point on infinity (refer to the Curve Points Encoding section for more details).
+  - <ins>Output:</ins> 192 bytes represent one point $\in E'(F_{p^2})$ in its decompressed form. In case of an empty input, it outputs the point at infinity (refer to the Curve Points Encoding section for more details).
 - ERROR_CODE = 1:
   - Points are incorrectly encoded (refer to Curve points encoded section).
   - Point is not on the curve.
@@ -772,10 +772,10 @@ pub fn bls12381_g2_multiexp(
 
 ***Description:***
 
-The function takes the list of field elements $a_i \in F_p$ and maps them to $G_1 \subset E(F_p)$. 
+This function takes as input a list of field elements $a_i \in F_p$ and maps them to $G_1 \subset E(F_p)$. 
 You can find the specification of this mapping function in the section titled 'Map to curve specification.' 
-Importantly, this function does NOT perform the mapping of the byte string into $F_p$. 
-The implementation of this function may vary and can be effectively executed within the contract.
+Importantly, this function does NOT perform the mapping of the byte string into $F_p$.
+The implementation of the mapping to $F_p$ may vary and can be effectively executed within the contract.
 
 ***Input:*** 
 
@@ -832,7 +832,9 @@ pub fn bls12381_map_fp_to_g1(
 
 ***Description:***
 
-The function takes the list of elements $a_i \in F_{p^2}$ and maps them to $G_2 \subset E'(F_{p^2})$. You can find the mapping function's specifications in the "Map to Curve Specification" section. It's important to note that this function does NOT map the byte string into $F_{p^2}$. The implementation of this function may vary and can be effectively executed within the contract.
+This function takes as input a list of elements $a_i \in F_{p^2}$ and maps them to $G_2 \subset E'(F_{p^2})$.
+You can find the mapping function specification in the "Map to Curve Specification" section. It's important to note that this function does NOT map byte strings into $F_{p^2}$.
+The implementation of the mapping to $F_{p^2}$ may vary and can be effectively executed within the contract.
 
 ***Input:*** the function takes as input `96*k` bytes — the elements from $F_{p^2}$ (two unsigned integers $< p$). Additional details can be found in the Curve Points Encoding section.
 
@@ -851,7 +853,7 @@ The ERROR_CODE is returned.
 - Validate the results for known answers from EIP-2537[^47],[^48]
 - Generate a random point $a$ from $F_{p^2}$:
   - Verify the result with another library.
-  - Check that result point on curve in $G_2$.
+  - Check that the resulting point lies in $G_2$.
   - Compare results for $a$ and $-a$; they should have the same x-coordinates and opposite y-coordinates.
 
 Edge cases:
@@ -900,13 +902,13 @@ $e([a]P, [b]Q) = e(P, Q)^{ab} = e([b]P, [a]Q)$
 
 This function is necessary to verify BLS signatures/zkSNARKs.
 
-This function takes as input the sequence of pairs $(p_i, q_i)$, where $p_i \in G_1 \subset E(F_{p})$ and $q_i \in G_2 \subset E'(F_{p^2})$ and validate:
+This function takes as input the sequence of pairs $(p_i, q_i)$, where $p_i \in G_1 \subset E(F_{p})$ and $q_i \in G_2 \subset E'(F_{p^2})$ and validates:
 
 $$
 \prod e(p_i, q_i) = 1
 $$
 
-We don’t calculate the pairing function itself: the result will be in the huge field, and in all known applications only this validation check is necessary.
+We don’t need to calculate the pairing function itself as the result would lie on a huge field, and in all known applications only this validation check is necessary.
 
 ***Input:*** A sequence of pairs $(p_i, q_i)$, where $p_i \in G_1 \subset E(F_{p})$ and $q_i \in G_2 \subset E'(F_{p^2})$. Each point is encoded in decompressed form. An expected input size of 288*k bytes is anticipated, interpreted as byte concatenation of k slices. Each slice comprises the concatenation of an uncompressed point from $G_1 \subset E(F_p)$ (occupying 96 bytes) and a point from $G_2 \subset E'(F_{p^2})$ (occupying 192 bytes). Additional details can be found in the Curve Points Encoding section.
 
@@ -962,7 +964,7 @@ For an empty input, the function returns ERROR_CODE = 0.
 - The first point is not on the curve.
 - The second point is not on the curve.
 - Input length exceeds the memory limit.
-- Incorrect encoding of the infinity point.
+- Incorrect encoding of the point at infinity.
 - Incorrect encoding of a curve point:
   - Incorrect decompression bit.
   - Coordinates greater than or equal to 'p'.
@@ -1016,7 +1018,7 @@ The ERROR_CODE is returned.
 - The input is beyond memory bounds. 
 - Point is not on the curve. 
 - Incorrect decompression bit. 
-- Incorrectly encoded infinity point. 
+- Incorrectly encoded point at infinity.
 - Point with a coordinate larger than 'p'.
 
 ***Annotation:***
@@ -1059,7 +1061,7 @@ pub fn bls12381_decompress_g2(&mut self,
 
 #### General comments for all functions
 
-In all functions, input is fetched from memory, beginning at `value_ptr` and extending to `value_ptr + value_len`. 
+In all functions, the input is fetched from memory, beginning at `value_ptr` and extending to `value_ptr + value_len`. 
 If `value_len` is `u64::MAX`, input will come from the register.
 
 Execution ends only if there's an incorrect input length,