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Here’s the public key generated by the private key we created above, shown as the co‐ ordinates x and y. Public Key K defined as a point K = (x,y). x = F028892BAD...DC341A 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 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 transac‐ tions per day, adds a significant amount of data to the blockchain. As we saw in the section “Public Keys” on page 65 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 y2 mod p = (x3 + 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 y2, 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 com pressed public key with the prefix 02 if the y is even, and 03 if it is odd, allowing the 78 | Chapter 4: Keys, Addresses, WalletsPDF Image | Mastering Bitcoin
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