@ -135,11 +135,11 @@ People can pay for goods and services using bitcoin as the currency. mg
Bitcoin transactions, which transfer value from one bitcoin address to another, are recorded in a distributed ledger, called the _blockchain_. In simple terms, think of the ledger as a book with lines like this:
----
Address 27 gave 2 bitcoin to address 81
Address 132 gave 1.05 bitcoin to address 22
25 bitcoin were mined to address 76
Address 13 gave 0.5 bitcoin to address 52
Address 52 gave 0.015 bitcoin to address 166
- Address 27 gave 2 bitcoin to address 81
- Address 132 gave 1.05 bitcoin to address 22
- 25 bitcoin were mined to address 76
- Address 13 gave 0.5 bitcoin to address 52
- Address 52 gave 0.015 bitcoin to address 166
----
The ledger is a record of all bitcoin transactions and can be independently verified by every node.
@ -39,32 +39,33 @@ Elliptic curve multiplication can be visualized on a curve as drawing a line con
.Elliptic Curve Cryptography: Visualizing the addition operator on the points of an elliptic curve
image::images/ecc-addition.png["Addition operator on points of an elliptic curve"]
Bitcoin specifically uses the +secp256k1+ elliptic curve which is a standardized curve on a group field of large prime order:
Bitcoin specifically uses the +secp256k1+ elliptic curve:
[latexmath]
++++
\begin{equation}
{y^2 = (x^3 + 7)} over \mathbb{F}_p
\end{equation}
++++
or
[latexmath]
++++
\begin{equation}
{y^2 \mod p = (x^3 + 7) \mod p}
where p = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F, a very large prime.
\end{equation}
++++
where +p = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F+, a very large prime.
The +mod p+ indicates that this curve is over a finite field of prime order +p+, also written as F(p). The curve looks like a pattern of dots scattered in two dimensions, which makes it difficult to visualize. However, the math is identical as that of an elliptic curve over the real numbers shown above.
[[ecc-over-F37-math]]
.Elliptic Curve Cryptography: Visualizing the addition operator on the points of an elliptic curve over F(p)
image::images/ecc-over-F37-math.png["Addition operator on points of an elliptic curve over F(p)"]
[TIP]
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The bitcoin private key is just a number. A public key can be generated from any private key. Therefore, a public key can be generated from any number, up to 256-bits long. You can pick your keys randomly using a method as simple as dice, pencil and paper.
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==== Generating bitcoin keys
@ -72,10 +73,11 @@ The first and most important step in generating keys is to find a secure source
[TIP]
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The size of bitcoin's private key, 2^256^ is a truly unfathomable number. It is equal to approximately 10^77^ in decimal. The visible universe contains approximately 10^80^ atoms.
The bitcoin private key is just a number. A public key can be generated from any private key. Therefore, a public key can be generated from any number, up to 256-bits long. You can pick your keys randomly using a method as simple as dice, pencil and paper.
====
[[privkey_gen]]
.Private key generation: From random mouse movements to a 256-bit number used as the private key
<1> Multiplying the priv_key by the generator point of the elliptic curve group, produces the pub_key
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[TIP]
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The size of bitcoin's private key, 2^256^ is a truly unfathomable number. It is equal to approximately 10^77^ in decimal. The visible universe contains approximately 10^80^ atoms.
Andreas M. Antonopoulos
11 years ago