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nadams 2017-04-24 08:59:23 -07:00
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@ -472,7 +472,7 @@ 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!
Whereas uncompressed public keys have a prefix of +04+, ((("Wallet Import Format (WIF)","for compressed keys")))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 earlier. 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. Public key compression is illustrated in <<pubkey_compression>>.
Whereas uncompressed public keys have a prefix of +04+, ((("Wallet Import Format (WIF)","for compressed keys")))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 earlier. 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. Public key compression is illustrated in <<pubkey_compression>>.
[[pubkey_compression]]
.Public key compression