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drusselloctal@gmail.com 2014-10-30 09:20:28 -07:00
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@ -389,15 +389,14 @@ KxFC1jmwwCoACiCAWZ3eXa96mBM6tb3TYzGmf6YwgdGWZgawvrtJ
The resulting WIF-compressed format starts with a "K". This denotes that the private key within has a suffix of "01" and will be used to produce compressed public keys only (see <<comp_pub>>).
===== Public Key Formats
===== Public key formats
Public keys are also presented in different ways, most importantly as either _compressed_ or _uncompressed_ public keys.
As we saw previously, the public key is a point on the elliptic curve consisting of a pair of coordinates +(x,y)+. It is usually presented with the prefix +04+ followed by two 256-bit numbers, one for the x-coordinate of the point, the other for the y-coordinate. The prefix +04+ is used to distinguish uncompressed public keys from compressed public keys that begin with a +02+ or a +03+.
As we saw previously, the public key is a point on the elliptic curve consisting of a pair of coordinates +(x,y)+. It is usually presented with the prefix +04+ followed by two 256-bit numbers, one for the _x_ coordinate of the point, the other for the _y_ coordinate. The prefix +04+ is used to distinguish uncompressed public keys from compressed public keys that begin with a +02+ or a +03+.
Here's the public key generated by the private key we created above, shown as the coordinates +x+ and +y+.
Here's the public key generated by the private key we created earlier, shown as the coordinates +x+ and +y+:
.Public Key K defined as a point +K = (x,y)+
----
x = F028892BAD...DC341A
y = 07CF33DA18...505BDB
@ -405,19 +404,18 @@ y = 07CF33DA18...505BDB
Here's the same public key shown as a 520-bit number (130 hex digits) with the prefix +04+ followed by +x+ and then +y+ coordinates, as +04 x y+:
.Uncompressed Public Key K shown in hex (130 hex digits) as +04xy+
----
K = 04F028892BAD...505BDB
----
[[comp_pub]]
===== Compressed Public Keys
===== Compressed public keys
Compressed public keys were introduced to bitcoin to reduce the size of transactions and conserve disk space on nodes that store the bitcoin blockchain database. Most transactions include the public key, required to validate the owner's credentials and spend the bitcoin. Each public key requires 520 bits (prefix \+ x \+ y), which when multiplied by several hundred transactions per block, or tens of thousands of transactions per day, adds a significant amount of data to the blockchain.
As we saw in the section <<pubkey>> above, a public key is a point (x,y) on an elliptic curve. Since the curve expresses a mathematical function, a point on the curve represents a solution to the equation and therefore if we know the x-coordinate we can calculate the y-coordinate by solving the equation y^2^ mod p = (x^3^ + 7) mod p. That allows us to store only the x-coordinate of the public key point, omitting the y-coordinate and reducing the size of the key and the space required to store it by 256 bits. An almost 50% reduction in size in every transaction adds up to a lot of data saved over time!
As we saw in the section <<pubkey>>, a public key is a point (x,y) on an elliptic curve. Because the curve expresses a mathematical function, a point on the curve represents a solution to the equation and, therefore, if we know the _x_ coordinate we can calculate the _y_ coordinate by solving the equation y^2^ mod p = (x^3^ + 7) mod p. That allows us to store only the _x_ coordinate of the public key point, omitting the y-coordinate and reducing the size of the key and the space required to store it by 256 bits. An almost 50% reduction in size in every transaction adds up to a lot of data saved over time!
Whereas uncompressed public keys have a prefix of +04+, compressed public keys start with either a +02+ or a +03+ prefix. Let's look at why there are two possible prefixes: since the left side of the equation is y^2^, that means the solution for y is a square root, which can have a positive or negative value. Visually, this means that the resulting y-coordinate can be above the x-axis or below the x-axis. As you can see from the graph of the elliptic curve, the curve is symmetric, meaning it is reflected like a mirror by the x-axis. So, while we can omit the y-coordinate we have to store the _sign_ of y (positive or negative), or in other words we have to remember if it was above or below the x-axis, as each of those options represents a different point and a different public key. When calculating the elliptic curve in binary arithmetic on the finite field of prime order p, the y coordinate is either even or odd, which corresponds to the positive/negative sign as explained above. Therefore, to distinguish between the two possible values of y, we store a +compressed public key+ with the prefix +02+ if the +y+ is even, and +03+ if it is odd, allowing the software to correctly deduce the y-coordinate from the x-coordinate and uncompress the public key to the full coordinates of the point.
Whereas uncompressed public keys have a prefix of +04+, compressed public keys start with either a +02+ or a +03+ prefix. Let's look at why there are two possible prefixes: because the left side of the equation is y^2^, that means the solution for y is a square root, which can have a positive or negative value. Visually, this means that the resulting _y_ coordinate can be above the x-axis or below the x-axis. As you can see from the graph of the elliptic curve in Figure 4-2, the curve is symmetric, meaning it is reflected like a mirror by the x-axis. So, while we can omit the _y_ coordinate we have to store the _sign_ of y (positive or negative), or in other words, we have to remember if it was above or below the x-axis because each of those options represents a different point and a different public key. When calculating the elliptic curve in binary arithmetic on the finite field of prime order p, the _y_ coordinate is either even or odd, which corresponds to the positive/negative sign as explained above. Therefore, to distinguish between the two possible values of y, we store a +compressed public key+ with the prefix +02+ if the +y+ is even, and +03+ if it is odd, allowing the software to correctly deduce the y-coordinate from the x-coordinate and uncompress the public key to the full coordinates of the point.
[[pubkey_compression]]
.Public Key Compression