 First question comes from Carl. Carl, can you explain the cryptography employed in the Bitcoin network and digital signatures in a little more detail, please? Alright, Carl. Challenge accepted. I'll give it my best. The cryptography employed primarily in Bitcoin, and in fact, in almost all of the other cryptocurrencies based on a field of mathematics called elliptic curve cryptography. Elliptic curve cryptography has been chosen for this task because in the field of elliptic curve, basic arithmetic works almost the same as arithmetic in the real numbers or the integers, with one important catch. Addition and multiplication work exactly the same as normal arithmetic, but there is no subtraction or division. If you multiply two numbers together, you can easily get the product of those numbers, but if you try to divide, you cannot do that. The only way to do division is to try all possible factors, a brute force search, and that is the basis on which elliptic curve cryptography operates. This is effectively a one-way function, and this type of one-way function is used in many different ways in cryptography. You might be thinking, how is it possible to create a one-way function? I think one of the best examples I've seen, and you can see this online in a YouTube video, in fact, is comparing this to mixing paints. Let's say I take a blue color and a yellow color. Is it easy to mix those two colors together? Very easy to mix them together. I can mix them together, and it will produce some shade of green. If I show you that shade of green, can you tell exactly which two shades of blue and yellow I use to produce it? The truth is you can't. If you've ever tried to paint your house and gone to the store and said, can you make me the exact same color as I have on my walls, you will quickly discover that people have difficulty doing that. They don't know which paints you mixed exactly to produce that color. The elliptic curve works in a very similar way. Digital signatures use some of the tricks of addition and multiplication, knowing that division is impossible, to produce a proof that the person who has created the digital signature had possession of the private key, but without revealing that private key. When you produce a digital signature, you are generating a number. That number is the result of multiplying and adding together a random number and your private key, and producing a new number, which we can prove by comparing it to the public key. It was produced by the private key, but we don't know what the private key is. That little trick is how you do digital signatures. The digital signature allows you to prove that you know the private key, and anyone who has the public key can check that it is a signature produced related to that public key, so they know it's from that private key, but they don't know what the private key is. That proof of knowledge, knowing what the private key is without revealing the private key, is the essence of digital signatures. The specific algorithm that's used, called ECDSA, or Elliptic Curve Digital Signature Algorithm, is detailed in a bunch of different places. You can also find it in chapter six of Mastering Bitcoin. On the GitHub repository, for example, if you go and look under the section for ECDSA math, you will find a brief description of the formula that's used. Specifically, in order to create a digital signature, first we create a random number. We use that random number, which is called the ephemeral private key, to produce a point on the Elliptic Curve by multiplying it with a generator point. We then take the x-value of that, add to it our known private key, and multiply that by our known private key, and then we add that to the hash of the message, which produces a specific value that we use as part of our signature. It's better to look at the equation to understand how that works. If you don't understand the equation, it doesn't matter. The important thing to realize is this. A private key is a number, a public key is a point on the Elliptic Curve that's produced by multiplying, that private key number with a specific other point called the generator point. As long as you know the private key, you can create the public key. If you know the public key, you cannot figure out what the private key is. So that's how the security of this system works.