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drusselloctal@gmail.com 2014-10-30 09:22:31 -07:00
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@ -415,7 +415,7 @@ Compressed public keys were introduced to bitcoin to reduce the size of transact
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! 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: 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. 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 <<ecc-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 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]] [[pubkey_compression]]
.Public Key Compression .Public Key Compression