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@ -6,7 +6,7 @@ Before you can get started programming Bitcoin (and Lightning), you should have
Bitcoin is a programmatic system that allows for the transfer of the bitcoin currency. It is enabled by a decentralized, peer-to-peer system of nodes, which include full nodes, wallets, and miners. Working together, they ensure that bitcoin transactions are fast and non-repudiable. Thanks to the decentralized nature of the system, these transactions are also censor-resistant and can provide other advantages such as pseudonymity and non-correlation if used well.
Obviously, Bitcoin is the heart of this book, but it's also the originator of many other systems, including blockchains and Lightning, which are both detailed in this tutorial, and many other cryptocurrencies such as Ethereum and Litecoins, which are not.
Obviously, Bitcoin is the heart of this book, but it's also the originator of many other systems, including blockchains and Lightning, which are both detailed in this tutorial, and many other cryptocurrencies such as Ethereum and Litecoin, which are not.
**_How Are Coins Transferred?_** Bitcoin currency isn't physical coins. Instead it's an endless series of ownership reassignments. When one person sends coins to another, that transfer is stored as a transaction. It's the transaction that actually records the ownership of the money, not any token local to the owner's wallet or their machine.
@ -56,11 +56,11 @@ ECC does not receive much attention in this tutorial. That's because this tutori
**_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.
Any line that intersects an elliptic curve will do so at either 1 or 3 points ... and that's the basis of elliptic-curve cryptography.
**_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 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>` - 1`.
**_How is an Elliptic Curve Defined Over a Finite Field?_** An elliptic 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>` - 1`.
**_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.