Clean up for new sections on ECC and Lightning.

01_0_Introducing_Bitcoin.md

@@ -54,9 +54,9 @@ It's important to Bitcoin (and to most blockchain systems) because it's the basi
 
 **_What Is a Signature?_** A message (or more commonly, a hash of a message) can be signed with a private key, creating a signature. Anyone with the corresponding public key can then validate the signature, which verifies that the signer owns the private key associated with the public key in question. 
 
-**_What Is a Hash Function?_** A hash function is an algorithm frequently used with cryptography. It's a way to map a large, arbitrary amount of data to a small, fixed amount of data. Hash functions used in cryptography are one-way and collision-resistant, meaning that a hash can very reliably be linked to the original data, but the original data can not be regenerated from the hash. Hashes thus allow the transmission of small amounts of data to represent large amounts of data, which can be important for efficiency and storage requirements. 
+**_What Is a Hash Function?_** A hash function is an algorithm frequently used with cryptography. It's a way to map a large, arbitrary amount of data to a small, fixed amount of data. Hash functions used in cryptography are one-way and collision-resistant, meaning that a hash can reliably be linked to the original data, but the original data can not be regenerated from the hash. Hashes thus allow the transmission of small amounts of data to represent large amounts of data, which can be important for efficiency and storage requirements. 
 
-Hashes also disguise what the original data is, which in the case of Bitcoin allows the concealment of the actual public key, making transactions resistant to quantum computing.
+Bitcoin takes advantage of a hash's ability to disguise the original data, which allows concealment of a user's actual public key, making transactions resistant to quantum computing.
 
 ### Public-Key Cryptography — In Short
 
@@ -64,23 +64,23 @@ One way to think of public-key cryptography is: _a way for anyone to protect dat
 
 ## About ECC
 
-ECC stands for elliptic-curve cryptography. It's a specific branch of public-key cryptography that depends on mathematical calculations conducted using elliptic curves defined over a final field. It's more complex and harder to explain than classic public-key cryptography (which used prime numbers), but it has some nice advantages.
+ECC stands for elliptic-curve cryptography. It's a specific branch of public-key cryptography that depends on mathematical calculations conducted using elliptic curves defined over finite fields. It's more complex and harder to explain than classic public-key cryptography (which used prime numbers), but it has some nice advantages.
 
-ECC is not really focused on in this book. That's because this book is all about integrating with Bitcoin Core and Lightning servers, which have already taken care of the cryptography for you. In fact, this book's particular focused was developed so that you don't have to worry about cryptography, because that's something that you _really_ want to expert to deal with.
+ECC does not receive much attention in this tutorial. That's because this tutorial is all about integrating with Bitcoin Core and Lightning servers, which have already taken care of the cryptography for you. In fact, this tutorial's intention is that you don't have to worry about cryptography at all, because that's something that you _really_ want experts to deal with.
 
-**_What is an Elliptic Curve?_** An elliptic curve is a geometric curve that takes the form `y<sup>2</sup> = x<sup>3</sup> + ax + b`. A specific elliptic curve is chosen by the careful definition of `a` and `b`. For cryptographic purposes, it must then be carefully examined to determine if it works well for cryptography. For example, the secp256k1 used by Bitcoin is defined as `a=0` and `b=7`.
+**_What is an Elliptic Curve?_** An elliptic curve is a geometric curve that takes the form `y<sup>2</sup> = x<sup>3</sup> + ax + b`. A specific elliptic curve is chosen by selecting specific values of `a` and `b`. The curve must then be carefully examined to determine if it works well for cryptography. For example, the secp256k1 curve used by Bitcoin is defined as `a=0` and `b=7`.
 
 Any line that intersects an elliptic curve will do so at either 1 or 3 points ... and that's the basis of elliptic-curve cryptopgrahy.
 
 **_What are Finite Fields?_** A finite field is a finite set of numbers, where all addition, subtraction, multiplication, and division is defined so that it results in other numbers also in the same finite field. One simple way to create a finite field is through the use of a modulo function. 
 
-**_How is an Elliptic Curve Defined Over a Finite Field?_** An ellipitic curve defined over a finite field is a specifically defined elliptic curve where all the points are drawn from a specific finite field. It's effectively defined as `y<sup>2</sup> % field-size = (x<sup>3</sup> + ax + b) % field-size` The finite field used for secp256k1 is `2<sup>256</sup> - 2<sup>32</sup> - 2<sup>9</sup> - 2<sup>8</sup> - 2<sup>7</sup> - 2<sup>6</sup> - 2<sup>4</sup> - <sup>1</sup>`. 
+**_How is an Elliptic Curve Defined Over a Finite Field?_** An ellipitic curve defined over a finite field has all of the points on its curve drawn from a specific finite field. This takes the form: `y<sup>2</sup> % field-size = (x<sup>3</sup> + ax + b) % field-size` The finite field used for secp256k1 is `2<sup>256</sup> - 2<sup>32</sup> - 2<sup>9</sup> - 2<sup>8</sup> - 2<sup>7</sup> - 2<sup>6</sup> - 2<sup>4</sup> - <sup>1</sup>`. 
 
-**_How Are Elliptic Curves Used in Cryptography?_** In elliptic-curve cryptography, a user picks a very large (256-bit) number as his private key. He then adds a set base point on the curve to itself that many times. (In secp256k1, the base point is `G = 04 79BE667E F9DCBBAC 55A06295 CE870B07 029BFCDB 2DCE28D9 59F2815B 16F81798 483ADA77 26A3C465 5DA4FBFC 0E1108A8 FD17B448 A6855419 9C47D08F FB10D4B8`, which prefixes the two parts of the tuple with an `04` to say that it's in unomcpressed form.) The resultant number is the public key. Various mathematical formula can then be used to prove ownership of the public key, given the private key. As with any cryptographic function, this one is a trap door: it's easy to go from private key to public key and largely impossible to go from public key to private key.
+**_How Are Elliptic Curves Used in Cryptography?_** In elliptic-curve cryptography, a user selects a very large (256-bit) number as his private key. He then adds a set base point on the curve to itself that many times. (In secp256k1, the base point is `G = 04 79BE667E F9DCBBAC 55A06295 CE870B07 029BFCDB 2DCE28D9 59F2815B 16F81798 483ADA77 26A3C465 5DA4FBFC 0E1108A8 FD17B448 A6855419 9C47D08F FB10D4B8`, which prefixes the two parts of the tuple with an `04` to say that the data point is in uncompressed form. If you prefer a straight geometric definition, it's the point "0x79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798,0x483ADA7726A3C4655DA4FBFC0E1108A8FD17B448A68554199C47D08FFB10D4B8") The resultant number is the public key. Various mathematical formula can then be used to prove ownership of the public key, given the private key. As with any cryptographic function, this one is a trap door: it's easy to go from private key to public key and largely impossible to go from public key to private key.
 
-This particular methodology also explains why finite fields are used in elliptic curves: it ensures that the private key will not grow too large. Note that the finite field for secp256k1 is slightly smaller than 256 bits, which means that all public keys will be 256 bits large, just like the private keys are.
+This particular methodology also explains why finite fields are used in elliptic curves: it ensures that the private key will not grow too large. Note that the finite field for secp256k1 is slightly smaller than 256 bits, which means that all public keys will be 256 bits long, just like the private keys are.
 
-**_What Are the Advantages of ECC?_** The main advantage of ECC is that it allows the same security as classic public-key cryptography with a much smaller key. A 256-bit elliptic-curve public key corresponds to a traditional 3072-bit public key.
+**_What Are the Advantages of ECC?_** The main advantage of ECC is that it allows the same security as classic public-key cryptography with a much smaller key. A 256-bit elliptic-curve public key corresponds to a 3072-bit traditional (RSA) public key.
 
 ### ECC - In Short
 
@@ -133,29 +133,29 @@ Do note that there may still be solutions for some of these situations within th
 
 ## About Lightning
 
-Lightning is a layer-2 protocol that interacts with Bitcoin to allow users to use their bitcoins "off-chain". It has both advantages and disadvantages over using Bitcoin on its own.
+Lightning is a layer-2 protocol that interacts with Bitcoin to allow users to exchange their bitcoins "off-chain". It has both advantages and disadvantages over using Bitcoin on its own.
 
-Lightning is also the secondary focus of this book. Though the book is mostly about interacting directly with Bitcoin (and the `bitcoind`), Lightning is an upcoming technology that is likely to become a popular alternative to Bitcoin in the near future. This book thus takes the same approach to Lightning as to Bitcoin: it teaches how to interact directly with a trusted Lightning daemon from the command line.
+Lightning is also the secondary focus of this tutorial. Though it's mostly about interacting directly with Bitcoin (and the `bitcoind`), it pays some attention to Lightning because it's an upcoming technology that is likely to become a popular alternative to Bitcoin in the near future. This book takes the same approach to Lightning as to Bitcoin: it teaches how to interact directly with a trusted Lightning daemon from the command line.
 
 Unlike with Bitcoin, there are actually several variants of Lightning. This tutorial uses the standard-compliant [c-lightning](https://github.com/ElementsProject/lightning) implementation as its trusted Lightning server.
 
 **_What is a Layer-2 Protocol?_** A layer-2 Bitcoin protocol works on top of Bitcoin. In this case, Lightning works atop Bitcoin, interacting with it through smart contracts.
 
-**_What is a Lightning Channel?_** A Lightning Channel is a connection between two Lightning users. Each of the users locks up some number of bitcoins with a multi-sig signed by both of them. The two users can then exchange bitcoins without ever writing to the blockchain. Only when they want to close out their channel do their settle their bitcoins, based on the final division of coins.
+**_What is a Lightning Channel?_** A Lightning Channel is a connection between two Lightning users. Each of the users locks up some number of bitcoins on the Bitcoin blockchain using a multi-sig signed by both of them. The two users can then exchange bitcoins through their Lightning channel without ever writing to the Bitcoin blockchain. Only when they want to close out their channel do they settle their bitcoins, based on the final division of coins.
 
-**_What is a Lightning Network?_** When you put together a number of Lightning Channels, this creates the Lightning Network. This allows two users who have not created a channel between themselves to exchange bitcoins with Lightning. The protocol forms a chain of Channels between the two users, then exchanges the coins through the chain using time-locked transactions.
+**_What is a Lightning Network?_** Putting together a number of Lightning Channels creates the Lightning Network. This allows two users who have not created a channel between themselves to exchange bitcoins using Lightning: the protocol forms a chain of Channels between the two users, then exchanges the coins through the chain using time-locked transactions.
 
-**_What are the Advantages of Lightning?_** Lightning allows for faster transactions with lower fees. This creates the real possibility for bitcoin-funded micropayments. It also offers better privacy, since it's off-chain with only the first and last states of the transaction being written to the immutable Bitcoin ledger. 
+**_What are the Advantages of Lightning?_** Lightning allows for faster transactions with lower fees. This creates the real possibility of bitcoin-funded micropayments. It also offers better privacy, since it's off-chain with only the first and last states of the transaction being written to the immutable Bitcoin ledger. 
 
-**_What are the Disadvatnages of Lightning?_** Lightning is still a very new technology and hasn't been tested as thoroughly as Bitcoin. That's not just a question of the technological implementation, but also whether the design itself can be gamed in any unexpected ways.
+**_What are the Disadvantages of Lightning?_** Lightning is still a very new technology and hasn't been tested as thoroughly as Bitcoin. That's not just a question of the technological implementation, but also whether the design itself can be gamed in any unexpected ways.
 
 ### Lightning - In Short
 
-One way to think of Lightning is: _a way to transact bitcoins using electronic contracts between pairs of people, so that only a first and final state have to be written to the blockchain_.
+One way to think of Lightning is: _a way to transact bitcoins using off-chain channels between pairs of people, so that only a first and final state have to be written to the blockchain_.
 
 ## Summary: Introducing Bitcoin
 
-Bitcoin is a peer-to-peer system that allows for the transfer of funds through transactions that are locked with puzzles. These puzzles are dependent upon public-key cryptography. When you generalize the ideas behind Bitcoin, you get blockchains, a technology that's currently growing and innovating.
+Bitcoin is a peer-to-peer system that allows for the transfer of funds through transactions that are locked with puzzles. These puzzles are dependent upon public-key elliptic-curve cryptography. When you generalize the ideas behind Bitcoin, you get blockchains, a technology that's currently growing and innovating. When you expand the ideas behind Bitcoin, you get layer-2 protocols such as Lightning, which expand the currency's potential.
 
 ## What's Next?