Fix Typos in Documentation Files

barretenberg/cpp/src/barretenberg/ecc/fields/field_docs.md

@@ -11,7 +11,7 @@ We use Montgomery reduction to speed up field multiplication. For an original el
 The goal of using Montgomery form is to avoid heavy division modulo \f$p\f$. To compute a representative of element $$c = a⋅b\ mod\ p$$ we compute $$c⋅R = (a⋅R)⋅(b⋅R) / R\ mod\ p$$, but we use an efficient division trick to avoid straight modular division. Let's look into the standard 4⋅64 case:
 1. First, we compute the value $$c_r=c⋅R⋅R = aR⋅bR$$ in integers and get a value with 8 64-bit limbs
 2. Then we take the lowest limb of \f$c_r\f$ (\f$c_r[0]\f$) and multiply it by a special value $$r_{inv} = -1 ⋅ p^{-1}\ mod\  2^{64}$$ As a result we get $$k = r_{inv}⋅ c_r[0]\ mod\ 2^{64}$$
-3. Next we update \f$c_r\f$ in integers by adding a value \f$k⋅p\f$: $$c_r += k⋅p$$ You might notice that the value of \f$c_r\ mod\ p\f$ hasn't changed, since we've added a multiple of the modulus. A the same time, if we look at the expression modulo \f$2^{64}\f$: $$c_r + k⋅p = c_r + c_r⋅r_{inv}⋅p = c_r + c_r⋅ (-1)⋅p^{-1}⋅p = c_r - c_r = 0\ mod\ 2^{64}$$ The result is equivalent modulo \f$p\f$, but we zeroed out the lowest limb
+3. Next we update \f$c_r\f$ in integers by adding a value \f$k⋅p\f$: $$c_r += k⋅p$$ You might notice that the value of \f$c_r\ mod\ p\f$ hasn't changed, since we've added a multiple of the modulus. At the same time, if we look at the expression modulo \f$2^{64}\f$: $$c_r + k⋅p = c_r + c_r⋅r_{inv}⋅p = c_r + c_r⋅ (-1)⋅p^{-1}⋅p = c_r - c_r = 0\ mod\ 2^{64}$$ The result is equivalent modulo \f$p\f$, but we zeroed out the lowest limb
 4. We perform the same operation for \f$c_r[1]\f$, but instead of adding \f$k⋅p\f$, we add \f$2^{64}⋅k⋅p\f$. In the implementation, instead of adding \f$k⋅ p\f$ to limbs of \f$c_r\f$ starting with zero, we just start with limb 1. This ensures that \f$c_r[1]=0\f$. We then perform the same operation for 2 more limbs.
 5. At this stage we are left with a version of \f$c_r\f$ where the first 4 limbs of the total 8 limbs are zero. So if we treat the 4 high limbs as a separate integer \f$c_{r.high}\f$, $$c_r = c_{r.high}⋅2^{256}=c_{r.high}⋅R\ mod\ p \Rightarrow c_{r.high} = c\cdot R\ mod\ p$$ and we can get the evaluation simply by taking the 4 high limbs of \f$c_r\f$.
 6. The previous step has reduced the intermediate value of \f$cR\f$ to range \f$[0,2p)\f$, so we must check if it is more than \f$p\f$ and subtract the modulus once if it overflows.

barretenberg/cpp/src/barretenberg/ecc/pippenger.md

@@ -24,7 +24,7 @@ For example, let's say that our bit slice is 6 bits. The first round will take t
 
 So, for example, if the most significant 6 bits of a scalar are `011001` (25), we add the scalar's point into the 25th bucket. 
 
-At the end of each round, we then 'concatenate' all of the buckets into a sum. Let's represent each bucket accumulator in an array `A[num_buckets]`. The concatenation phase will compute `A[0] + 2A[1] + 3A[2] + 4A[3] + 5A[4] + ... = Sum`.
+At the end of each round, we then 'concatenate' all the buckets into a sum. Let's represent each bucket accumulator in an array `A[num_buckets]`. The concatenation phase will compute `A[0] + 2A[1] + 3A[2] + 4A[3] + 5A[4] + ... = Sum`.
 
 Finally, we add each `Sum` point into an overall accumulator. For example, for a set of 254 bit scalars, if we evaluate the most 6 significant bits of each scalar and accumulate the resulting point into `Sum`, we actually need `(2^{248}).Sum` to accommodate for the bit shift. 
 

docs/docs/aztec/concepts/accounts/keys.md

@@ -111,7 +111,7 @@ In the following section we describe a few ways how an account contract could be
 
 #### Using a private note
 
-Storing the signing public key in a private note makes it accessible from the entrypoint function, which is required to be a private function, and allows for rotating the key when needed. However, keep in mind that reading a private note requires nullifying it to ensure it is up to date, so each transaction you send will destroy and recreate the public key. This has the side effect of enforcing a strict ordering across all transactions, since each transaction will refer the instantiation of the private note from the previous one.
+Storing the signing public key in a private note makes it accessible from the entrypoint function, which is required to be a private function, and allows for rotating the key when needed. However, keep in mind that reading a private note requires nullifying it to ensure it is up-to-date, so each transaction you send will destroy and recreate the public key. This has the side effect of enforcing a strict ordering across all transactions, since each transaction will refer the instantiation of the private note from the previous one.
 
 #### Using an immutable private note
 

docs/docs/protocol-specs/addresses-and-keys/precompiles.md

@@ -8,7 +8,7 @@ Precompiled contracts, which borrow their name from Ethereum's, are contracts no
 
 Note that, unlike user-defined contracts, the address of a precompiled [contract instance](../contract-deployment/instances.md) and the [identifier of its class](../contract-deployment/classes.md#class-identifier) both have no known preimage.
 
-The rationale for precompiled contracts is to provide a set of vetted primitives for [note encryption](../private-message-delivery/private-msg-delivery.md#encryption-and-decryption) and [tagging](../private-message-delivery/private-msg-delivery.md#note-tagging) that applications can use safely. These primitives are guaranteed to be always-satisfiable when called with valid arguments. This allows account contracts to choose their preferred method of encryption and tagging from any primitive in this set, and application contracts to call into them without the risk of calling into a untrusted code, which could potentially halt the execution flow via an unsatisfiable constraint. Furthermore, by exposing these primitives in a reserved set of well-known addresses, applications can be forward-compatible and incorporate new encryption and tagging methods as accounts opt into them.
+The rationale for precompiled contracts is to provide a set of vetted primitives for [note encryption](../private-message-delivery/private-msg-delivery.md#encryption-and-decryption) and [tagging](../private-message-delivery/private-msg-delivery.md#note-tagging) that applications can use safely. These primitives are guaranteed to be always-satisfiable when called with valid arguments. This allows account contracts to choose their preferred method of encryption and tagging from any primitive in this set, and application contracts to call into them without the risk of calling into an untrusted code, which could potentially halt the execution flow via an unsatisfiable constraint. Furthermore, by exposing these primitives in a reserved set of well-known addresses, applications can be forward-compatible and incorporate new encryption and tagging methods as accounts opt into them.
 
 ## Constants
 

docs/docs/protocol-specs/calls/public-private-messaging.md

@@ -15,7 +15,7 @@ Private functions are executed locally by the user, so that the user can ensure
 
 Given this natural flow from private-land to public-land, private functions can enqueue calls to public functions. But the opposite direction is not true. We'll see [below](#public-to-private-messaging) that public functions cannot "call" private functions, but rather they must pass messages.
 
-Since private functions execute first, they cannot 'wait' on the results of any of their calls to public functions.
+Since private functions execute first, they cannot 'wait' on the results of their calls to public functions.
 
 By way of example, suppose a function makes a call to a public function, and then to a private function. The public function will not be executed immediately, but will instead be enqueued for the sequencer to execute later.
 

docs/docs/protocol-specs/decentralization/p2p-network.md

@@ -61,7 +61,7 @@ When new participants join the network for the first time, they will need to loc
 
 Whilst the DiscV5 specification is still under development, the protocol is currently in use by Ethereum's consensus layer with 100,000s of participants. Nodes maintain a DHT routing table of Ethereum Node Records (ENRs), periodically flushing nodes that are no longer responsive and searching for new nodes by requesting records from their neighbours.
 
-Neighbours in this sense are not necessarily in geographical proximity. Node distance is defined as the bitwise XOR of the nodes 32 bit IDs.
+Neighbours in this sense are not necessarily in geographical proximity. Node distance is defined as the bitwise XOR of the nodes 32-bit IDs.
 
 ```
 distance(Id1, Id2) = Id1 XOR Id2

docs/docs/protocol-specs/intro.md

@@ -22,7 +22,7 @@ Some of the info we need to populate this document might have already been writt
 
 ## Diagrams
 
-To increase the probability of diagrams being up to date we encourage you to write them in `mermaid`. Mermaid is a markdown-like language for generating diagrams and is supported by Docusaurus, so it will be rendered automatically for you.
+To increase the probability of diagrams being up-to-date we encourage you to write them in `mermaid`. Mermaid is a markdown-like language for generating diagrams and is supported by Docusaurus, so it will be rendered automatically for you.
 You simply create a codeblock specifying the language as `mermaid` and write your diagram in the codeblock. For example:
 
 ````txt
@@ -87,7 +87,7 @@ classDiagram
 
 If mermaid doesn't cover your case, please add both the rendered image and the source code to the documentation. Most of the tools for diagramming can export a non-rendered representation that can then be updated by other people. Please name it such that it is clear what tool was used.
 
-This should allow us to keep the diagrams up to date, by allowing others to update them.
+This should allow us to keep the diagrams up-to-date, by allowing others to update them.
 
 ## For each protocol feature
 

docs/docs/protocol-specs/public-vm/alu.md

@@ -4,7 +4,7 @@ The algebraic logic unit performs operations analogous to an arithmetic unit in
 
 This component of the VM circuit evaluates both base-2 arithmetic operations and prime-field operation. It takes its input/output from the intermediate registers in the state controller.
 
-The following block diagram maps out an draft of the internal components of the "ALU"
+The following block diagram maps out a draft of the internal components of the "ALU"
 
 ![](/img/protocol-specs/public-vm/alu.png)
 

docs/docs/reference/developer_references/common_errors/aztecnr-errors.md

@@ -72,4 +72,4 @@ To address the error, register the account by calling `server.registerAccount(..
 
 You may encounter this error when trying to send a transaction that is using an invalid contract. The contract may compile without errors and you only encounter this when sending the transaction.
 
-This error may arise when function parameters are not properly formatted, when trying to "double-spend" a note, or it may indicate that there is a bug deeper in the stack (e.g. a bug in the Aztec.nr library or deeper). If you hit this error, double check your contract implementation, but also consider [opening an issue (GitHub link)](https://github.com/AztecProtocol/aztec-packages/issues/new).
+This error may arise when function parameters are not properly formatted, when trying to "double-spend" a note, or it may indicate that there is a bug deeper in the stack (e.g. a bug in the Aztec.nr library or deeper). If you hit this error, double-check your contract implementation, but also consider [opening an issue (GitHub link)](https://github.com/AztecProtocol/aztec-packages/issues/new).