Fix typos

Author: rex4539

docs/notes.md

@@ -614,7 +614,7 @@ The Bulletproofs paper outlines two versions of multi-party computation aggregat
 
 For more information on how the aggregation protocol works and is implemented, see the [protocol notes](../aggregated_range_proof/index.html). 
 
-The aggregated range proof has the same form as the individual range proof, in that the provers (the parties) still perform the same calculations to prove that \\(t(x) = \langle \mathbf{l}(x), \mathbf{r}(x) \rangle \\) and that \\(t_0, \mathbf{l}(x), \mathbf{r}(x)\\) are correct. The difference is that the challenge values are obtained from the dealer, which generates them by combining commitments from all the parties, and that the calculations of different parties are seperated by different powers of the challenge scalars \\(y\\) and \\(z\\).
+The aggregated range proof has the same form as the individual range proof, in that the provers (the parties) still perform the same calculations to prove that \\(t(x) = \langle \mathbf{l}(x), \mathbf{r}(x) \rangle \\) and that \\(t_0, \mathbf{l}(x), \mathbf{r}(x)\\) are correct. The difference is that the challenge values are obtained from the dealer, which generates them by combining commitments from all the parties, and that the calculations of different parties are separated by different powers of the challenge scalars \\(y\\) and \\(z\\).
 
 We will explain how one piece of the aggregated proof is generated for party \\(j\\), and then will show how all of the pieces for all of the \\(m\\) parties can be combined into one aggregated proof.
 
@@ -879,4 +879,4 @@ With these observations, we can simplify the combined \\(m\\)-party statement ab
 \\]
 
 
-[bulletproofs_paper]: https://eprint.iacr.org/2017/1066.pdf
+[bulletproofs_paper]: https://eprint.iacr.org/2017/1066.pdf

docs/range-proof-protocol.md

@@ -128,7 +128,7 @@ The dealer creates the aggregated vector polynomials \\(\mathbf{l}(x), \mathbf{r
 \end{aligned}
 \\]
 
-The the dealer then performs the [inner product argument](../inner_product_proof/index.html) to prove the relation:
+The dealer then performs the [inner product argument](../inner_product_proof/index.html) to prove the relation:
 \\[
 \operatorname{PK}\left\\{
   ({\mathbf{G}}, {\mathbf{H}}' \in {\mathbb G}^{n \cdot m}, P', Q \in {\mathbb G}; {\mathbf{l}}, {\mathbf{r}} \in {\mathbb Z\_p}^{n\cdot m})
@@ -289,4 +289,4 @@ multiplying equation `2` by \\(c\\), and adding it with equation `3`. Finally, t
                       + & \quad {\langle {- \mathbf{l}\_{(j)}(x)} -z\mathbf{1}, {\mathbf{G}\_{(j)}} \rangle}\\\\
                       + & \quad {\langle {- \mathbf{r}\_{(j)}(x)} \circ \mathbf{y}^{-n} y^{-j \cdot n} + z\mathbf{1} + z^{j+2} \cdot \mathbf{y}^{-n} y^{-j \cdot n} \circ {\mathbf{2}}^n, {\mathbf{H}}\_{(j)} \rangle}\\\\
 \end{aligned}
-\\]
+\\]

src/util.rs

@@ -105,7 +105,7 @@ pub fn scalar_exp_vartime(x: &Scalar, mut n: u64) -> Scalar {
 }
 
 /// Takes the sum of all the powers of `x`, up to `n`
-/// If `n` is a power of 2, it uses the efficient algorithm with `2*lg n` multiplcations and additions.
+/// If `n` is a power of 2, it uses the efficient algorithm with `2*lg n` multiplications and additions.
 /// If `n` is not a power of 2, it uses the slow algorithm with `n` multiplications and additions.
 /// In the Bulletproofs case, all calls to `sum_of_powers` should have `n` as a power of 2.
 pub fn sum_of_powers(x: &Scalar, n: usize) -> Scalar {