In this episode, Average Gary and Fundamentals dive into the fascinating world of mathematics, exploring its critical role in understanding Bitcoin and cryptography. They begin by discussing the motivation behind learning math, especially in the context of Bitcoin, and why it is essential for achieving personal sovereignty and a deeper understanding of cryptographic principles.
The hosts explore the common phrase "do the math" often heard in Bitcoin circles, unraveling the complexities behind it and emphasizing the importance of understanding the mathematical foundations of Bitcoin's protocol. They delve into the concept of finite fields, elliptic curves, and the significance of cryptographic algorithms, breaking down these complex topics into more digestible concepts.
Average Gary shares his journey into the world of math, driven by a desire to understand Bitcoin at a deeper level, and how this led him to explore cryptography and number theory. Fundamentals adds insights into the historical context of cryptography, its evolution, and its current applications in the Bitcoin space.
The episode also touches on the importance of self-learning and the availability of resources today that make it easier to study math and cryptography independently. They discuss the potential vulnerabilities in the Bitcoin ecosystem due to a lack of widespread understanding of cryptography and the need for more education in this area.
Listeners are invited to join this journey of exploration, as the hosts aim to demystify math and cryptography, empowering individuals to gain a higher level of understanding and confidence in their reasoning and decision-making related to Bitcoin and beyond.
[00:00:00]
Unknown:
We're gonna surf the wave. We are gonna surf the wave. I'm Average Gary. I'm here joined with
[00:00:07] Unknown:
Fundamentals. What's up, everybody?
[00:00:10] Unknown:
We're gonna talk about math. Oh, yes. Why is, why is yeah. So, like, buckle up. This is not your standard Bitcoin podcast. This is a math podcast. And I think first, we're gonna hash out, no pun intended, why? Like, why why does why is math why are we motivating the math here? Why is it important? Why is it applicable? What's motivating us specifically to dig into this? And then we'll we'll get into, like, the first, the first fundamentals of math that we need to learn if we're gonna accomplish these goals.
[00:00:46] Unknown:
Yes. So so why math? Yeah. Well, you know, we are, so we're we're connected by Bitcoin, 1st and foremost, probably. Right? I think we share a lot of interests, but, like, a lot like, I think in the world a lot of people share a lot of interest, but if they if the thing you share is a sort of a Bitcoin ethos, it's not like, it won't be the only thing you talk about, and it might not even be 90% of what you talk about, but it's probably, you know, it's maybe an unsaid thing in the background, like a daemon process, that you're probably interested in freedom, sovereignty, you know, and the like. Right? And, you know, there's a lot of I think there's a lot of kind of on the surface talk about math in Bitcoin.
It's usually related to, like, I hear do the math. I hear that as like a, you know Yeah. Saying,
[00:01:49] Unknown:
do the math. Right? As if, you know How do you calculate all these numbers that are going up? All these numbers, all these metrics Yeah. Right, are going up. How do you calculate it? Why does it matter? When you tell somebody like me to do the math, right, and then I go on to the GitHub and I see,
[00:02:04] Unknown:
like, a multi hundred character number as like a generator. I'm like, okay, well if I'm gonna do the math, I'm gonna have to memorize that number. So like people don't realize what they're saying when they say do the math. Right? They're saying understand that, you know, Bitcoin runs on a protocol, and that there it's math behind it, and 21,000,000 and all of that stuff. Right? But in order to really, I think, have a sovereign level of conviction. Right? From me personally, it required doing, a little more math than that. And I'm not there yet. I'm at I'm, like, 2 years into a rabbit hole of trying to achieve this.
[00:02:51] Unknown:
And we're gonna take a headlamp and we're gonna go down this rabbit hole. And Fundamentals are just gonna show us all his cave paintings along the way of this this beautiful rabbit hole. And and you touched on something there. It's like the sovereignty piece. Right? It's like the math is easy when you're looking at it from, like, your macro lens. Right? Like, that's usually everybody's like, oh, do the math. Right? It's like there's only gonna be 21,000,000. That's an easy equation to do. The math that we Divide by 2. Divide by 2. Years. That is what, I guess, maybe a lot of people
[00:03:18] Unknown:
say do the math. Right? Yeah. Divide by 2 every 4 years and, you know, maybe you learned what a geometric series was when you were in high school. I don't know. I don't know what that is. Yeah. Well, you probably do That's why we're here. That it's called that. Right? Yeah. That is why we're here. We're like there's probably a lot we do know, but you just don't remember what it's called or wherever. I you know, I never identified that it's called that. Yeah. There's a lot we know intuitively, and that's most of what we know intuitively is all we need to know, really.
[00:03:49] Unknown:
You know? Well, and the having like you said, that divide by 2 every 4 years or 210,000 blocks, that's, like, the the easy math. And you mentioned a word earlier, generator. And for those that are not attuned to what that is, or if you don't know what a generator is, that's a piece of a cryptographic algorithm. So when you're doing elliptic curve mathematics, there's a constant
[00:04:14] Unknown:
that's in there, and it's called a generator point. And I'm sure we'll get into, like, what that is. But, like There will be a moment where we we I literally describe what that is using only the numbers 1, 2, 3, 4, and 5 and basic multiplication. So, like, that's how can be learned at this very basic level. I think there are reasons why it's kept from us and nobody it's like, you know, it feels like the hardest thing in the world, and most people are so far away from being motivated to learn it. However, it is simple. An 8 year old, you know I actually have a book. Literally, it sits next to my where I podcast here just to remind me. And it's, like, literally a book about having to teach how to teach this stuff to 8 year olds. And it's this thin.
It's it's it's like, Less than an inch thick. Less than an inch thick. And, yeah, it's it's, his references a guy named Piaget who had a lot to say about gold back in, the 1800 as well. Very interesting. So but the bottom line is, you know, we're all gonna become children again. And that's actually how we're going to, that's how we're gonna navigate through this.
[00:05:22] Unknown:
Yeah. Well, the and the the understanding of math. Right? Because you can even if you learn how to code. Right? So I'm I'm a developer by trade. Like, I get paid to develop. And there there are many such people in the Bitcoin scene that are doing all these all these dev work and stuff. But then you have, like, this other this, like, next level class of Bitcoin citizens, we'll call them, which are, like, the cryptographers and the people doing the implementation, the the application of the cryptography and everything like that. And that's, like, the level, at least personally, that I strive to go to. Right? So I was Yeah. Previously working on, like, FEDIMENT and cashew. And I learned about, like, blinded signatures and, like, the way that equation worked out for, like, how that blinding function works, but the actual underlying math of the big numbers that are used for cryptography, like, the why of it. Right? It's more of, like, the why. Like, why are we using this generator? Why are we using these algorithms? Why does it work the way that it does?
Hopefully, we can kinda dig into it and dissect because that's the next level of understanding. Right? Right now, you as a Bitcoin, you know, if you're listening to your 40 hours of Bitcoin podcast, like you should be, you might understand the macro picture and everything like that. And maybe you understand some basics on how to run a node and how to do some development and everything like that. But that next level, in my opinion, the the the people that I have to trust are cryptographers. Right? So when I get this when I when I'm digging into this codebase, Fedimink codebase or whatever, and it's like, oh, yeah. This is, like, there's, like, this federated, like, consensus mechanism.
Right? And it uses, like, these signatures, these BLS signatures. I don't know what that means. I don't know why that works the way it does. And so when we got to chatting, we both kinda understood from a basic level that, like, there's a next level to understanding this Bitcoin thing. And that's the math behind it. Right? Who do you trust right now? I have to trust the cryptographers. You have to trust the cryptographers. Let's get to a level. Let's try to strive to get to a higher level of understanding
[00:07:22] Unknown:
so we don't have to trust those people as much. Or at least we can at least call b s on something. Absolutely. So let me describe real quick why I ended up in this rabbit hole. Because I am not a mathematician, and I was not I mean, there are people who would call me a financial mathematician as from my career or I was an actuary. But if you're an actuary, you know you're, like, not a mathematician. Right. You know you basically looked at the mountain of math, and then you looked at this job and said, I can handle this job without climbing that mountain. And then, you know, I have a I had a 30 year career where I had to visit the mountain every once in a while in order to do my job. And so I am an experienced self learner when I have to be. Mhmm. So but not you know, I never got a PhD in math or anything like that. But but I did end up in a rabbit hole that now has me look and sound very much the way, let's say, a mathematician in our space, right, asking the kinds of questions and more in in in that process. So let me just let me explain how I ended up here real quick. Yeah. So, like, I am 3 year I am 3 years from my entering Bitcoin, what I call my genesis moment. Alright?
And because of my experience, I guess I came up fairly fast. You know, I under I had a very good understanding of a lot of the sub subjects of bitcoin. I also, like, had a strong disdain for the knowledge system and, you know, so I was consuming bitcoin material. It was very it just felt very normal and it, you know. Yeah. It got I I came up fairly quickly and, one of the first things I found I'll I'll say 3 or 4 months in that captivated me was the base 58 curriculum. And so then I got really into that, you know, coding transactions and wanting to look at the GitHub and, you know, probably within 6 months I'd been to the git Bitcoin GitHub. I had a good grasp of, like, command line transactions.
I went through Jimmy Saum's programming Bitcoin book. And I had a pretty like, you go to the GitHub of Bitcoin, and you see how it works. You really can. Okay? And it's not you don't have to be a mathematician or even a coder to, like it's almost like it's written out in English. You know. So that's really that was really great. And then at some point, I can't remember which book. Like, there are a number of programming books that aren't widely talked about, but but they're really good about programming Bitcoin. I had this this thought that I had to go to, now the sec 256p1.
[00:10:09] Unknown:
P 256k1.
[00:10:11] Unknown:
Yeah. I had to go to that GitHub because now okay. I'm done relying on what the Bitcoiners did. Right? I understand what they did. I under you know, now it's time to understand what, Satoshi was relying on and what we're all relying on for that. And when I went to that GitHub, and this is, you know, like two and a half years ago, almost to the day, I saw I did not see a formula. I didn't see any variables or formulas. All I saw was just an unending sea of hard coded numbers, comma, the limited. And I was like, what the what the fucking hell is this? What is going on? How can anyone understand what the hell is going on here? Right? I thought it was gonna be like a road map, like Bitcoin is, like, you know, like the Bitcoin GitHub.
Yeah. Logic you can follow. That's what code is. Code is just logic that you can follow along. So, like, one of the things I used to be really strong at in my career was, if you gave me an Excel spreadsheet, even if it had 60 tabs in it, I would always be able to find where the the inputs come from and trace, you know, from beginning to end to understand that system. And so I you know, that this is how I that's my formula for understanding things. And the the the sec t 256k1 GitHub was just, like, it crushed me. Like, it literally, like, crushed my soul. It's like I was so I I was on this path. I consumed so much. I understood this bitcoin protocol.
And now I'm at this point of great reliance. Right? And I get nothing. I really get nothing. And there's nobody that can really tell me what to do. Right? There's certainly nobody in my network that I can ask Right. What do I do about this? Right? So here I am. Well, I mean, the answer the answer was, I guess, I need to go buy a cryptography book and start there. And I'm holding up I'm holding up the book I ended up buying, which I thought I think in retrospect was a home run. I'll just tell you. We'll put it in the show nights show notes. It's called Understanding Cryptography by Parim Peltzel is the author.
And one of the reasons I think it was a home run was, one, there's a lot of resources online related to this book. So, like, I'm pretty sure there's a lot of people who have taken taken cracks at all the quest, you know, solutions and you can see all that online. The authors put a lot online. And, it turns out I found out that there's a video seminar by the authors themselves, like, back when, like, open co open open courseware for MIT started. Yeah. I don't it wasn't part of that, but they did this. And, fair warning, if you do go watch them, most of the comments are telling them to get better cameras and that they're getting nauseous. But that's how early, like, they Yeah. This book was written in the late nineties. And so, I loved that it was before Bitcoin.
Right? And that's one of I like to read books that were before Bitcoin, especially on cryptography before Bitcoin, so it's not, you know, it's not super tainted by it and I have to, like, do more rewinding to sort the story out and their motives.
[00:13:40] Unknown:
Well, one of the things I remember on the intro in this book or one of the first chapters, because, I'm into chapter 2 now, but, like, in the first chapter it was talking about, like, sort of the history of cryptography or history of, cryptography. And it goes through, like, the general, like, there's cryptanalysis and all these other, like, different subsections of it. But it largely said that, hey, up until a certain point, cryptographers, like, the the mathematicians behind in doing the application of all this, were very much like almost like a state secret. Like, There was not this, like, open, open source, openly available course material or people out there talking about it. It just wasn't something like, if you were good at math and it led you to this, like, cryptography field, you're most definitely scooped up within the university system and put to work by, you know, some government somewhere. NSA, you'll be yeah. Something something like that. Right?
And only within the past, probably, 2 decades would be would would it has it really sort of sort of opened up. And I found that to be fascinating. And what that tells me too is well, first, let me say this. It's, like, whenever you open source information, which information wants to be free, right? When you when I share an idea with you, now there are 2 ideas that exist in the world, both in both of our brains. Right? So this inflationary nature of ideas and information, allows allows it to develop rapidly. And we see that in our modern world, like, information just proliferation everywhere. But if this idea of doing cryptography and and the math behind it and the techniques and everything like that, if it's only a couple decades old, just like many other things in our modern world, like, we're just starting to see, I think, the impacts of that. And Bitcoin might be one of the first largest impacts, let's say.
Right? I know Bitcoin was sort of the culmination of all these different technologies that came together to to create this this digital money. But what it's done now is it's intrigued and then also unlocked time for people that have the the time and the desire to, like, look into this idea that is cryptography, into this mathematical,
[00:15:48] Unknown:
you know, into these math equations. How does it work? Why does it work? That sort of thing. Yeah. And so something interesting is there's another thing. If you go back 20 years so there's another thing that didn't exist 20 years ago, which is you could not self study math. You didn't have access to even textbooks. So, like, the books that I have in my life, like, they weren't available. And when I told you before, like, there were times I did have to self study math in order to be able to, like, do my job. I would go, I live in Philadelphia area. I would go to the University of Pennsylvania bookstore, pretend I was a student, and go into the back where they had the textbooks for their students.
And that's the pretty much was the only way I was gonna be able to get a textbook to self learn. And MIT OpenCourse, software was really the first, I wanna say 15 year I mean, it was around 08, 09. It was, like, magically the same time that, Bitcoin was launched. You know, we also did have a financial crisis around that time too. And I think a lot of you know, like, whoever Satoshi was wasn't the only one who felt compelled to put something open source into the system for everybody at that moment. Right? Yeah. And, education started becoming much more, available open source and but it was unheard of. Like, you have to know it was absolutely unheard of. If you didn't have access to a university, you were not gonna get these books.
And, you know, you didn't have Amazon. You just didn't have I mean, you have, like, eBay. I don't I don't remember ever thinking I could get a textbook on eBay. So, like, it and so it's just one of those things that also the so cryptography, I mean, we have no idea the story of cryptography. We have no idea, like, what we're screwing with right now and even have trying to have this conversation. We have no idea what kind of secrets are still locked up in, you know, by the people you mentioned. Right? Mhmm. No no idea. But we do know that some we we we know that some of it, enough of it was leaked out. Right?
We're almost like it's like, you know, the MK Ultra experiments led to psychedelics leaking out and people having growth experiences as a result. Right? I mean, God knows why those things ever existed to begin with. But, cryptography, like in yeah, so you talk about the book. You know, they go back to, you know, they go back to Caesar time that, you know, they people love talking about how cryptography goes back to BC. Right? And it's it's an innate human desire to want to protect communication. Right? Yeah. So, this book does a great job of that. This book talks about like, I never heard of DSA I'm sorry, RSA or AES. I'd never heard of any of those.
I knew from my Bitcoin books now what a digital signature meant, what an elliptic curve meant. But, the book the other thing the book did that was really great was went through the math. Now, unfortunately, the math was foreign to me. So, that's what opened up the next rabbit hole. Right? It was like, oh, my God. I gotta now study number theory, abstract algebra. I mean, it felt gettable. It felt understandable. Like, you know, sometimes you look at a math subject and you're like, that's out of reach for the way my brain works. Right? Like, that's never good. I'm never gonna learn that. The the I started to see that the math needed to understand the wall of hard coded numbers was approachable within my lifetime.
[00:19:34] Unknown:
So, like, if I just got started getting to work, I might get some I might get somewhere. Right? It's sort of It's building the context around it. Right? It's like anything else. So you don't know until you can understand the surrounding context. So it's like, if you don't know the terminology, if you don't know the way things are structured, And this is applicable to basically anything. Right? This is one of the things, like, I was I was a linguist in in my past career in the Navy. And it's all about just building context. And so that's why, you know, immediately down the rabbit hole, I started listening to Bitcoin podcasts. I was talking about whatever. And you don't know anything when you start listening to these things. Right? And you hear these words and you're like, I don't I don't know. Like, when they start talking about, like, bond stuff or whatever. Like, I would hear the words and you can maybe understand the basic words.
But the the context of these words together have implications. Right? Like yield curve control. The first time I heard yield curve control on, like, a Bitcoin macro podcast, I was like, I don't know what that means. But I kept listening. And eventually, over time, you start to build the box around, and and you see the shape around these things and see the association. It's almost like you're building up, like, your LLM in your head. Right? Like, LLMs are just relations of of words to each other that occur in specific patterns that can be regurgitated. But human the human brain sort of works the same way where if you don't know the shape of the box that you're looking at or or the the the surrounding things, it's like the force for the trees type of deal. Right? You can stare at this one tree forever, but you kinda have to zoom out, zoom in and out at different times to understand what's going on around it. Like, what's adjacent to it? What's what do these words mean? Language is so imprecise in and of itself.
Context is critical.
[00:21:15] Unknown:
Yeah. And then so then where's the math fall into that? Is that so, like, when you get bombarded with some you learn something from somebody else. Right? Like, you read a book or you listen to a podcast. It takes a really long time to ever actually develop a thought for yourself, asking if this is right or wrong. Right? You're usually just gonna accept that it's right, especially if you read a book. They're like, oh, they wouldn't have printed it in this nice font if this was wrong. Right? You you read a cryptography textbook, you're going along to you're going to assume that it's right. You know, you're gonna and it takes I don't even know what it's almost spiritual at this moment where you start to say, what if this person might what if this is wrong? How do I know this is right?
How do I know I'm right? Right? Mhmm. And this is gonna sound strange, I think, to people who listen to this say this for the first time. But at a very deep level, for me, the learning math is the process of building conviction in your own reasoning. It's not just learning how to implement code. It's not just learning how to validate code. Right? I think most people can know that 10 plus 10 equals 20, and that's not but they don't ask themselves how they know. Like and there is a really deep question there.
[00:22:40] Unknown:
Yeah. Right?
[00:22:42] Unknown:
It's like, the great there's the man known as the greatest mathematician to ever live, his name is Gauss, Carl Friedrich Gauss. And in the I read his biography recently, one of the biographies, and it talks about how when he was 3 or 4 years old, he was adding large numbers. And there were no calculators to validate any of it. And the adults just thought it was cute. But if you ask him throughout his life, he always says he knew he was right. And, you know, you have to ask the question how. How does one know they're right? And when it comes to people who are relying on Bitcoin for their life savings and more, right, so, like, the longer you're in it, the more you're like, it's not just about my life savings, this is about my livelihood, Right? My livelihood is based on not caring, not depending on anybody for any aspect of my sovereignty, really. Right?
It's really important that you build a sense of conviction in your own self. And there's a lot of ways to do that. So you can, engage in self defense. Right? You can work out. You when you work out, you have an experience of yourself being strong. Right? And then that builds conviction, and then you work out more and better, and it becomes a cornerstone that you build on.
[00:24:09] Unknown:
Well, it exposes your weaknesses too. Like, when you go to work out and you hit something that you can't do.
[00:24:15] Unknown:
Yeah. And that's how you become strong. You break that muscle and rebuild it, and you give it the stimulus. You send it messages that you're not strong enough for this. So, you know, you're gonna have to get strong enough. So learning math and it's not the only way to build conviction in your reasoning. You you can study logic. You can do things. But, you know, it's really low hanging fruit to just get on, spend 15 minutes or 20 minutes a day learning math and filling in that part of your fitness. Because getting we this thing is so early. Right? This Bitcoin thing is very early. We went through a fork war in, you know, 2017, which I would argue is still is, you know, never ended.
But it's decided, but not certainly not settled. Right? That man, I think I'm really lucky that it was one of the first books I read when I got into the space. Because it's like the one book that I think has the most to do with my conviction is that one because it's the one that made me think what, really, what are the attack vectors? And when I started thinking about the math thing, so first, you know, yeah, I run into this GitHub, and it gives me nothing. So how do you so we have a we have now a society of Bitcoiners that can't validate cryptography. That's was my first thought. Right? Mhmm.
That was so when I say that I fell apart and I had a crisis I had, like, a personal crisis, it was like, oh my god. Who is minding the store here? Right? And then so is it up to me to start in this rabbit hole and try to answer this question? Because, otherwise, there are if there are people that do understand this, they're not sharing it publicly. They're not writing books to connect this dot to us. Right? The people writing books are connecting, you know, of the world are writing books on what they know. The people that can answer the question, what is that wall of numbers on the on the secp GitHub, I don't know that they're intentionally keeping secret. They may not think it's interesting to people, but they are not writing books about it. And it is a, to me, a glaring weakness in, our ability to be resilient to, you know, I'll say to a social attack.
And make no mistake, like, I started, the more I started getting exposed to, like, mathematicians and PhDs, I started, you know, hundreds of mathematicians whose YouTube sites I use, and then I go check out their biography. They all work for shit coins. It's like, wow. Fucking weird. Right? Yeah. That's weird that they all are, like they all want to do blockchain research and stuff like that. And then it's, like, no. It's not weird. That makes total sense. They're the that's who pays. Right? Yep. And then, you know, so then there's this small group of people that aren't doing that, like the Peter, you know, Peter Willis of the world. And but they all work for 1 company for the most part. Yeah.
And we don't have now, like Andrew Polstra, I've met him and I've found him to be very candid when I'm and ask answering questions when I meet him, but it's not like I can't ask him the question I had.
[00:27:40] Unknown:
You know? It's like you said earlier, like, people in your network like, you can bring these questions, but there's not anybody in your network to to tap into to get them answered or to know or to trust or to understand from.
[00:27:53] Unknown:
And that's Yeah. So I have this fear, like, what if, you know, what if one day the the guys we trust wanna set up a fork and they're they're convincing us that everything is cool, trust us. Maybe they can be trusted, but who could, you know, who could validate it?
[00:28:09] Unknown:
Right.
[00:28:10] Unknown:
Right? Who could validate it and tell everybody either it's cool or it's not? And that was what scared me. Right? Because I remember I read that book, Block Size Ward, and I was like, wow, people were pretty sure of themselves. It really impressed me. It was like, wow, people really must have understood Bitcoin really well to have understood that increasing that block size could have such an impact. But that was a non mathematic you know, they weren't using mathematical weakness against people. They were just using you know, this was like It's an economic thing. I would just say the attack was early and Yeah. Like lame. You know? And it should've worked anyway, and it didn't because peep you know, but, you know, it's a beautiful thing.
But to me, the attack the vulnerability of mostly, say, Bitcoiners who don't give a shit, like, don't give a shit about math
[00:29:04] Unknown:
is very much out there. Right? So where do we begin giving a shit? Right? Like, where what what where do we start?
[00:29:11] Unknown:
Where so I so I always I have 3 kinda key reasons. So I basically said, man, I think I need to start a math institute somehow. Right? We somehow, they're and somebody needs to put energy into making sure we get better at this. Yep. And I had 3, like like, 3 key things I thought would come out of there. 1 would was, like, personal sovereignty. Like, you can't have you can't be relying on somebody else to do your math. Right? It's like, okay. You're really not sovereign if someone's calculate you know, if someone's doing calculations for you. It's you know, my grandfather was accountant for people, and I just remember how much people relied on him. And I remember thinking, man, that must suck for those people to have to rely on somebody for their own business to know what's going on. Right? You can't be personally sovereign if you're asking, you know, understand there's too many incentives to to, you know, for that to go badly. And the other so, the second thing I thought about were homeschoolers and the rising tide of homeschoolers that have no conviction in their ability to teach their kids math. And, like, what are they gonna do when their kids start showing a lot of talent Yep. And it freaks them out. And then, like, are they gonna break and send them to a quote unquote real school because they don't feel like they can do the job adequately at home? Or they they lack they don't know where to go. Right? They lack the resources. Something like what we're trying to build doesn't exist.
And they don't know where to go to even, you know, to even know what to do. Right? So I saw that as a bit. And then and then I saw the just the fork possibility and just, like, like, god. Everyone just needs to level up, like, a little bit, and we will be a lot stronger.
[00:30:56] Unknown:
Yeah. Well, one of the recent, you know, topics du jour in the Bitcoin ecosystem, which I don't think was gonna go away, it's sort of an evergreen topic, is this whole idea of, like, quantum resistant things. Right? And and if so somebody is proposing some new signature schema. For instance, I think last year, sometime last year, the NIST, NIST, it's like the National Institute of Standards and Technology or something, released these, like, candidates or selected candidates or something for post quantum secure digital signature algorithms. Right? Yeah. So the post quantum I went to go look at them. Yes. This is quantum attack. When I went to go look at them, but, like, I don't I don't know how to, like, how do you know, I could probably go through the meeting notes or something like that or some resource of, like, how they determine these are the ones.
But without understanding math, the math fundamentals, the math basics Yeah. Of how all this shit works,
[00:31:50] Unknown:
I I'm I'm looking at Greek. Right? I'm reading Greek and and it doesn't make sense. Know what the problem you're trying to solve is. So I can tell you, when I read this book, Understanding Cryptography, I don't know if it was chapter 1 or chapter 2. It was very early on, I became very comfortable that at least with the fact that it was not a difficult exercise to raise the hashing out, you know, to raise the hashing out, though. If you wanna go from shot 256 to shot 512, it's not it's like a pain in the ass. It's not it's not an, you know, it's not a game over. It's a pain in the ass. Right. And, I became comfortable. So I think, you know, like and I think learning in a lot of ways, we're dealing with time machines here. K? So, like, a hashing algorithm is like an anti time machine. Like, it will take you literally forever to the end of time unless you know the one thing you need to know to
[00:32:43] Unknown:
Find the solution. To undo it. You would just will never, you know, it'll never happen. You can't you can't muster enough power. That's why I say These are called one way functions. Right? Like, hashing algorithms that were or generally referred to as a one way function. It's like a smoothie. You put bananas, strawberries, yogurt into a blender, and out the other side, you get a strawberry banana smoothie. Right? But you can't take that smoothie and go back to the exact banana, the exact strawberries, the exact amount of yogurt. Right? You just can't you can't do that. That's right. So Hashing Algae is
[00:33:14] Unknown:
largely that same way. Now there's a a very simple mathematical concept, which is, you know, you just look at both directions and what it's gonna require. You just can you can see what calculations, what bit operations are required to go one way versus the other way. And that's, you know, that's something I think we wanna demystify. And I so there's a certain level of things that just need demystifying. That will make a big difference. Then there's this level of things that are, like, they're kinda difficult, but they're so cool that they're worth learning. And then there's things that are difficult to the point where we're we are just trying to touch it. We don't know it yet. So I would I would classify most of, like, what we wanna cover in those three categories. Most of it, I think, is demystifying.
Okay. And once it's demystified because that's the thing. Like, what gets weaponized against you is, like, your infinite sense of what this could be. And you you're you're in this Dunning Kruger. You know the Dunning Kruger effect. Right? Where, like, people who don't know what the hell they're talking about can sound really smart because they're not stopped by their even their own incompetence. Right? And so they talk very confidently. And people who are competent see them as, oh, shit. Maybe they actually are onto something. And I don't know anything about this. So this is an infinite world of trouble for me. And we wanna take that infinite world of trouble, turn it into a finite world of trouble, and then and then make it smaller and smaller and smaller to the point where it's it's like, well, no. This is something I can handle. It's like, you know, imagine you your personal finances, you never did anything before. You never touched them. You never looked at it. And now, all of a sudden, you have an event in your family or something, and you have to now handle it. It feels infinite. You're like, oh my god. This is just an infinite thing. And then you crack you you crack it every day.
A little bit Clarifying the context. Right? A little bit every day. Yeah. And then it becomes a smaller and smaller world to the point where I mean, this is probably if if anyone here knows, like, understands cryptography, that's definitely how that worked. Like, when you first started, it seemed infinite. Right? Yeah. So we wanna do that with the math side because that's the thing that probably affects the most amount of people. There's probably more people in the world that just are intimidated unnecessarily to the point where they lose power. They cede power over it. Right? Right. So that's what we wanna, like, demystify a lot of this so that you no longer cede power over not knowing it. And then you can start actually understanding and build build a conviction for yourself.
So so what's the first thing? You'd mentioned a few before we started recording. What what, like, what was that first thing on your list? It was a list of things that we were gonna go over today as, like, the entry point. So there's a thing there are the things today, which is, like, the the the, you know, what most books would call section 0. But, like, when I talked about a general curriculum, I think when I was I was saying general number theory, we're talking division algorithm, Euclidean algorithm, Chinese remainder theorem, and, you know, then like finite fields. And these all, like, it's so weird. Like, my daughter is a math major. And she would be intimidated by those words for the most part, you know, because they don't you don't even learn stuff you don't learn that stuff until late.
You know, math majors learn calculus and how they can apply it to physics for the most part, for, like, the majority of their math majorhood. And then if they survive all that, they get to learn the kind of cool stuff that, you know. So that may sound that may sound intimidating. Right? But, like, the division algorithm is basically just how you divide numbers. And, it's like, if you remember when you were in 3rd grade, you did long division and you basically said, okay, I have something I have a factor that does multiply this number and then I have a remainder. And the algorithm is basically through a repeated process of finding remainders, you end up solving this division.
And you get into this thing called greatest common divisor and find that numbers that are relatively prime to each other, meaning that they have no greatest common divisor. These are important concepts in cryptography to understand, eventually understanding public key cryptography and very basic, like, discrete log type concepts.
[00:37:57] Unknown:
Yeah. Because when you divide it, it was like, I remember the the little drawing. Right? It was almost like, like a curved vertical line and then a straight line. And you're dividing the thing on the right side underneath this little drawing by the thing on the left. Yep. And if the thing on the right was, like, 100 and the thing on left was 3, it's, like, okay. Let's figure out, like, how many times 3 goes into a100. And then you write the answer on the top, and maybe there's a remainder. Right? That's right. Sometimes there's a remainder of 0.
[00:38:27] Unknown:
Those are nice. Right? We consider those nice. Right? Right. And then when there's a remainder that's non zero, we're like, ugh.
[00:38:34] Unknown:
But that's critical for cryptography, though. Yes. Because that remainder math
[00:38:39] Unknown:
is one of the things that I think we'll get into, in fact. Yeah. One of the things I was thinking about for section 0, we didn't even it was not even on my list, was just talking about the different number systems that we know about and exist, like the integers, the rationals, the reals, because there's a system of remainders, Right? That which is what we end up in a with in a prima in a finite field, called residues. System of residues modulo a prime. And, which, if anybody has opened, like, Programming Bitcoin by Jimmy Song and did that first three pages Yep. You you have a prime number like 5, and then so this this number system of the residues, modulo that 5, would just be 1, 2, 3, 4.
[00:39:29] Unknown:
And It's 0.
[00:39:31] Unknown:
Well, right. But then when you, and 0 is implied. Yeah. It's called an equivalence class. But, like, that's a number system just like all the integers. Right? But that's a number system that possesses the very special properties of like closure. Meaning, like, if you add any two numbers in that system, 2+4 is 6, but 6 modulo 5 is 1 and that's in your system. That's in your 1, 2, 3, 4. So, like, you can't do that with the integers. You can't do that with the rationals or the real numbers. You can only do that in this system of remainders.
[00:40:06] Unknown:
Right. So naming I guess, naming these, like, groups of numbers and stuff like that, you can you can, again, contextualize. You can put a box around what this is. So if we're talking about, what are they called? They're called sets or fields? Or
[00:40:19] Unknown:
So fields is a very specific structure. Mhmm. Right? And a finite field is an even more specific structure. It's a field that has a finite amount of elements. This happens to be the thing we care about in the cryptography we use around Bitcoin.
[00:40:36] Unknown:
So in your example, though, of this this finite field, I think it's referred to of the order of 5. Right? Yes. Yes. So finite field of the order of 5 is 01234.
[00:40:51] Unknown:
Yeah. But we don't we but those actually, the 0 is not the like, the 0 doesn't go in the field. It's not it's it's it's, it's not part of that finite field necessarily, because it's already it's already, like, it's already equivalent. The remainder of 0s is already equivalent. Not to get too not to get, you know, not to get caught up in a, you know, in a detail. What matters is that if you multiply any two numbers in those fields, you get a number in that field. If you add any two numbers in that field, you get a number in that field. And, yes, a 0 could be, you know, 0 could be a 0 could be one of those numbers.
Right. Right? You could add 4 and 1, and you get 0 because it's the remain remember, we're talking remain the the the members of this field are remainders after dividing by 5. Right?
[00:41:46] Unknown:
So, like whatever the operation is. So if you're doing 4 plus 1, no matter what you do, you do that normally. Right? So 4+1 would be 5, but 5 is not within it. We've exceeded the bounds of this finite field. And just no matter what you do, you always do if you're if you're exceeding the bounds of the finite field, you do a modulo
[00:42:07] Unknown:
Yeah.
[00:42:08] Unknown:
Or a division, and then keep the remainder. The modulo just means divide by this number, and then whatever the remainder is is the answer.
[00:42:17] Unknown:
Yes. So, like, it's it's another number system, like the integers. Right? It's that's and I don't think like, for me, I spent 30 years, like, learning a lot of math, and I got to this thing about finite fields, and I almost shut the book after 2 pages because I just, like, couldn't, like, I couldn't rock it. I was, like, what the heck is like, there's no there's no system where if I multiply two numbers, it is in the field. Like, that's crazy.
[00:42:48] Unknown:
Well, clock math. This is clock math. Yes. I think colloquial, we can refer to as clock math. Yes. If you have if I've had 2 o'clock, 2 PM, 1400 well, we'll use 1400 for military time. Yes. And you add 56 hours to it. Yes. Right? You don't add 14 plus 56, which would be, like, 70. It's not 70 o'clock. You have to once you reach the end of that finite field of numbers, which is 24 hours in a day, you sort of loop back to the beginning again. Yes. Great example. You keep doing that over and over to stay within the bounds of it. But the quickest way to do that would be, you know, 14 +56modulo24 is sort of the rep way you could represent it, which means 14 +56, which is 70 divided by 24.
And then whatever the remainder is is what you're at. Yep. Is is the answer. Which I think is 22. Well, I was just about to punch it in. I don't know.
[00:43:50] Unknown:
And I yeah. I got that because 7 it I know 72 is a multiple of 24. So negative 2 +24 would be 22. Let's find out. I don't know. Let's find out. Big test on fundamentals, mental math here. Please be right.
[00:44:11] Unknown:
56 hours. Hold on. Alright. Let's say say it out loud again.
[00:44:15] Unknown:
You had 4 you were at 14 o'clock. 14 o'clock. You had a 56 hours. Yep. Right. So you got to 70 o'clock. But there's no such thing as 70 o'clock. We only go up to 24.
[00:44:25] Unknown:
Right? So we divide by 24.
[00:44:27] Unknown:
And you probably get a remainder of 2. So you got right? You got 48. You can account for 48 of those hours of those 70 hours. Right? And then what's left is 22 is what's left is 22.
[00:44:41] Unknown:
Yeah. So 70 minuteus 48
[00:44:43] Unknown:
is 22. Yep. That's that's your that's your new remainder. Right? Mod in a modulo 24
[00:44:51] Unknown:
work. So 2 PM, 1400 plus 56 hours is is 10 PM. 10 PM. Yep.
[00:45:00] Unknown:
Cool.
[00:45:02] Unknown:
And that same principle is how these elliptic curves work. Correct? Very similar. Yes. And,
[00:45:10] Unknown:
probably there's a good I mean, at some point we can get into why visually this makes sense. But, like, if you graphed a ellipse, right? You graphed an elliptic function in real like with the real numbers, you would see a drawing that looks like an oval. So, it's pretty easy to look at the oval and find what points made what points gave you that oval. Very easy to do with the real numbers. Right? You see the ellipse, that oval shape, and you know the formula. Right? It's something like x squared over a plus y squared over b equals something. And you're like, okay, I know it's an ellipse centered around, a point. Right? And I can figure out exactly where the input is.
But when you locate it in a clock math type field, it becomes a scattered thing that you can't locate the origin of. That's why it's used. And that is why these things are used in cryptography. That, like, and in the in the book Programming Bitcoin and in the I'm pretty sure in the book Mastering Bitcoin by Antonopoulos Mhmm. There's some really great visual examples of why,
[00:46:30] Unknown:
you know, of how that works. Right? And and the equation that we specifically are talking about, this this secp256k1equation, is y squared is equal to x cubed plus 7. Yeah. That is, like, that is the definitive question. So if you go on, you know, like, some Wolfram Alpha, you know, Matlievs type of site that you can plot these equations, If you just punch that equation into there, it will plot it for you, and it'll it'll it'll give you a nice illustration.
[00:47:01] Unknown:
And you could but you can be in 7th grade. You could see a picture of that elliptic function, really. Right? Mhmm. And you could say to yourself, yeah. I have a pretty good idea of what the x I I have a pretty good idea of what the x and y are. Right? Yeah. I can kinda visualize it pretty well, but when you plot that and this is a this is going to be hard to explain and visualize. It's just if when you plot we'll we'll put a picture of this in the show notes from that book. But when you plot it over this numbers not a real number system, but a finite field number system, Modulo a certain prime, which you can go on bit get Bitcoin's GitHub and see what that prime is. It's gigantic.
Right? But if you, in the book, they'll they do it and they'll say, oh, let's, if the prime was like 31, something very manageable, you can see. It's, it looks like a, it looks like a random scatter plot is the bottom line. It's when you plot it over a finite field, it looks like a random scatterplot. So when you look at a point on that field, you have no clue where it came from.
[00:48:11] Unknown:
So it's almost like these these numbers, these these groupings of numbers. Right? Like, real numbers. I don't know which one it represents. Yeah. When we're using real numbers, these equations take a a recognizable shape. So they're they're predictable in a sense because when we plot them out, there is a shape, and and, you know, humans are great at shape, pattern recognition, and everything like that. And so if you apply these finite fields using these prime numbers to this shape, it completely obliterates the shape to the visual eye. There are still patterns. Like, it's still following a set equation.
But the it it it's almost like it's, it's like obfuscating what the true shape of this group of numbers is, even though there is a pattern to it. Yeah. So it's a bit it's it's a visual
[00:49:00] Unknown:
explanation of why these things are why it's cryptographically, you know, why is it used in this way? Right? How how is this used to obfuscate the origin, right, the origin of the point? Like, at some point, we can we'll get talk about public key cryptography. The whole point is I can tell you a point on the curve, right, without telling you where the origin was. Right? That's the whole kinda that's the big premise. It's the beauty of it. Right? I can I can tell a partner I wanna work with what my, what the answer is? And then if he can guess how the answer was created, then he can, you know, he can move something.
Right? He can access this he can access something that I have to offer him. Right? It at some point, you have to be able to put out there something in public. Right? Yeah. That can't be backed into. And so this concept and this is why, like, we talked just section 0 a little bit about the number systems. Right? Because you it's if you understand, a little bit about there are these number systems. 1 is, like, the integers, very simple, like, we call whole numbers, like 1, 2, 3, 4, also negative 1, negative 2, negative 3. Right? Those it's just those. Right? And then you have this other number system called rationals, which is like, these are now, you can make fractions of integers.
What we call P over Q, where P and Q are integers. So you have 1 half, 1 eighth, onesixteenth. Right? 3 28ths. Those are now rational systems. So you have a little bit more number. You have more numbers than in the integer system. Right? And with more usefulness. For example, the integers, you can't take it back, you can't invert an integer that's not 1. You can't, in other words, you can't multiply an integer by another integer that gives you what's called the identity, I. E. 1. You need a rational number for that. So if I had the number 3 and I wanted to get to the number 1 by multiplication, I need 1 third.
So we created the rational numbers to be able to do that. Right? And then, there's, between the rational, from the rational numbers, you have a big leap to what's called the real numbers, which is, now, that includes numbers that can't be represented as a fraction, like the square root of 2, like pi, things of those natures. And there are a lot of those numbers. That's a, I mean, it is a field in and of itself understanding the real numbers. But with the real numbers, we get to see things like continuity. We get to see continuity between the numbers, which we never really could see in rational numbers.
[00:51:59] Unknown:
What do you mean by continuity?
[00:52:01] Unknown:
In other words, do you ever draw a function, like a function, and you can see how the line doesn't break as you move across the x axis. The numbers the numbers represented on the y axis by the function do not break. Whereas, if you all if all you had were integers, you would have dots. If you had rational numbers, you'd have more dots, a lot more dots, but you'd have a lot of gaps. In the real numbers, basically, we say the rest is are the gaps. So that we can now have things like continuity, which gives us calculus, you know, gives us optimization, gives us all derivatives and integrals, setting derivatives equal to 0 and finding maximums and minimums. Like all of that is because of continuity and we get to do that because of the real number system. That we get that to because we get to imagine that it's complete.
Right? We get to imagine that there's absolute completion. Right? So that system was created really to fulfill that. You know? Somehow, we had this innate ability to think about it, but without the number system to work with. Right?
[00:53:14] Unknown:
But in numbers, we we we use numbers to, like, sort of try to represent the real world. Right? Like, the this this idea of, like, what is this table? Right? There's there's you could break this table down into the mathematical formulas that constitute, like, the atoms that create it and everything like that. Like, I feel like the more we try to explain the world with numbers
[00:53:37] Unknown:
Mhmm.
[00:53:38] Unknown:
And having these different sets of numbers or or, to to use is a way for explaining different aspects of the world. When you're talking about the the the real numbers and, like, pi being one of them, my mind goes to, like, oh, that's why pi is part of, like, a circle. Right? Like, how do you represent the idea of a complete circle?
[00:53:57] Unknown:
Right? Yeah. Like and so and it's funny because if you started with, well, I thought let's say I started with a square or triangle. Right? Or square or Mhmm. A poly you know, and I I say, okay. Well, I have a square, and I see how a square is kinda like a circle. Right? If I didn't have a circle, I'd probably just say, oh, well, then just give me a square. Right? Right. And then because I can make a square at 4 sides, 90 degree angles. Right? Now, okay, what if I had a pentagon or a hexagon or a decagon? You see, the more sides I add The closer you get to a circle. The closer I get to the circle. But you have to imagine this idea that I'm gonna have infinitely many sides in order to actually have a circle.
[00:54:45] Unknown:
Yeah. Yeah. It's the it's like the granularity. You see that sometimes, in, like, image displays. Right? Like the the poly polygons, the number of polygons that go into an image. This is something like video games and stuff. It's like the number of polygons. Like, PS PlayStation 1 had, like, low polygon resolution. Right? It's why, like, Lara Croft looked like a box when she was running around. Right? But in the modern days Money for Nothing video. What's
[00:55:11] Unknown:
that? The there's a video for a song called Money for Nothing.
[00:55:14] Unknown:
Yeah.
[00:55:15] Unknown:
That's I highly recommend everybody go check that out on YouTube. It's, iconic song from the eighties by Dire Straits. And, they really used at the time cutting edge CGI. But if you look at it now, it's it looks like the most primitive thing ever.
[00:55:34] Unknown:
Yeah. So, I guess, it's it's we're further we're getting further resolution in in in by using these different numbers. And so where do where do where do finite fields fit in? Is that its own It's another number system. So it's another type of number system
[00:55:50] Unknown:
to answer a different question. So, like, the real numbers were You might say the real numbers were created to answer how do I solve a question called, x squared minus 2 equals 0? That's know, at which which, Pathagoras asked. Right? Mhmm. That's a question from the days of Pathagoras. And it's a there's a really interesting story that, you know, so, Pythagoras was the king of the right triangle. Right? Mhmm. And the theorems the theorem his theorem, the a squared plus b squared equals c squared, which were the two sides of a right triangle, if you squared them, they became the square of the, what's called the hypotenuse? The hypotenuse. Yeah. The long side. Well, if you have, if your legs are 1 unit, which people love to think about anything using a unit of 1. So what if you had your sides 1 unit?
Right? How do you represent the long edge of that triangle? And, you know, so Pythagoras was a he was a bit of an influencer in his time. Okay. And he had a group there were a group of people called the Pythagoreans who followed every all those teachings. And, there was we they knew what rational numbers were. They didn't really know what irrational numbers were. They just knew that they they knew there were numbers that couldn't be represented by well, at least they thought there had to be, possibly, numbers that couldn't be represented by fractions. Right? Mhmm. But, Pythagoras completely rejected this idea.
Pythagoras thought absolutely the existence of an irrational number is absurd. And, one of his students basically said, yeah. But what if your what if one of your triangles, you know, has a long edge square root of 2? That's, you know, square root of 2 is irrational. And you know what happened to that guy? His name was like Hep Hepha Hepha sis. Hepha sis. You know what happened to him? A bunch of the Pythagoreans took him on a boat and dumped him in the ocean, killed him. They didn't they actually killed him And, because this was blasphemy.
Yeah. And they didn't well, it wasn't I mean, and, you know, part of it was like if the world figures out there are irrational numbers and this whole Pythagoras thing is bunk. And guess what? The one of the most, like, popular proofs there is in the world, and people love coming up with different ways to do it, is to prove that square root of 2 is irrational. And it's almost like it's it's a fuck you to the the idea of the Pythagoreans. Right? Right. Slay your this guy's slayed his hero. And, you know, so we, you know, so, you know, we standing at the base of cryptography, we're not that different from, like, the Pythagoreans because we're we're staring into a space we know very little about. We're actually asking, do these things exist? Right?
So they were asking, does the is there even a number system that includes these numbers? And, you know, the answer was yes. And it was controversial. It was so contra you know, it was so controversial. It. And so now the question is, is there a number system that allow that really facilitates, what we know as cryptography and the ability to protect value in a secret. Right? And the answer is yes. And it's called the it's it has a name in English called the finite field, and I feel like that cheapens it. I think that the thing to the thing of the people to get is that there is a number system, right, that's similar to the number systems we know of, but not exactly like it has its own structure and rules, but there is a number system that makes it possible.
[00:59:59] Unknown:
Man, I think we I think we cut it there. I don't know. That's a that's a beautiful introduction to, I think, what we're gonna explore here. Cool. Yeah. And, for anybody listening in, thank you, 1st and foremost, for giving us one of your most scarce resources. And I look forward to going down this rabbit hole and looking at all the cave drawings that you've painted, Fundamentals. Nice.
We're gonna surf the wave. We are gonna surf the wave. I'm Average Gary. I'm here joined with
[00:00:07] Unknown:
Fundamentals. What's up, everybody?
[00:00:10] Unknown:
We're gonna talk about math. Oh, yes. Why is, why is yeah. So, like, buckle up. This is not your standard Bitcoin podcast. This is a math podcast. And I think first, we're gonna hash out, no pun intended, why? Like, why why does why is math why are we motivating the math here? Why is it important? Why is it applicable? What's motivating us specifically to dig into this? And then we'll we'll get into, like, the first, the first fundamentals of math that we need to learn if we're gonna accomplish these goals.
[00:00:46] Unknown:
Yes. So so why math? Yeah. Well, you know, we are, so we're we're connected by Bitcoin, 1st and foremost, probably. Right? I think we share a lot of interests, but, like, a lot like, I think in the world a lot of people share a lot of interest, but if they if the thing you share is a sort of a Bitcoin ethos, it's not like, it won't be the only thing you talk about, and it might not even be 90% of what you talk about, but it's probably, you know, it's maybe an unsaid thing in the background, like a daemon process, that you're probably interested in freedom, sovereignty, you know, and the like. Right? And, you know, there's a lot of I think there's a lot of kind of on the surface talk about math in Bitcoin.
It's usually related to, like, I hear do the math. I hear that as like a, you know Yeah. Saying,
[00:01:49] Unknown:
do the math. Right? As if, you know How do you calculate all these numbers that are going up? All these numbers, all these metrics Yeah. Right, are going up. How do you calculate it? Why does it matter? When you tell somebody like me to do the math, right, and then I go on to the GitHub and I see,
[00:02:04] Unknown:
like, a multi hundred character number as like a generator. I'm like, okay, well if I'm gonna do the math, I'm gonna have to memorize that number. So like people don't realize what they're saying when they say do the math. Right? They're saying understand that, you know, Bitcoin runs on a protocol, and that there it's math behind it, and 21,000,000 and all of that stuff. Right? But in order to really, I think, have a sovereign level of conviction. Right? From me personally, it required doing, a little more math than that. And I'm not there yet. I'm at I'm, like, 2 years into a rabbit hole of trying to achieve this.
[00:02:51] Unknown:
And we're gonna take a headlamp and we're gonna go down this rabbit hole. And Fundamentals are just gonna show us all his cave paintings along the way of this this beautiful rabbit hole. And and you touched on something there. It's like the sovereignty piece. Right? It's like the math is easy when you're looking at it from, like, your macro lens. Right? Like, that's usually everybody's like, oh, do the math. Right? It's like there's only gonna be 21,000,000. That's an easy equation to do. The math that we Divide by 2. Divide by 2. Years. That is what, I guess, maybe a lot of people
[00:03:18] Unknown:
say do the math. Right? Yeah. Divide by 2 every 4 years and, you know, maybe you learned what a geometric series was when you were in high school. I don't know. I don't know what that is. Yeah. Well, you probably do That's why we're here. That it's called that. Right? Yeah. That is why we're here. We're like there's probably a lot we do know, but you just don't remember what it's called or wherever. I you know, I never identified that it's called that. Yeah. There's a lot we know intuitively, and that's most of what we know intuitively is all we need to know, really.
[00:03:49] Unknown:
You know? Well, and the having like you said, that divide by 2 every 4 years or 210,000 blocks, that's, like, the the easy math. And you mentioned a word earlier, generator. And for those that are not attuned to what that is, or if you don't know what a generator is, that's a piece of a cryptographic algorithm. So when you're doing elliptic curve mathematics, there's a constant
[00:04:14] Unknown:
that's in there, and it's called a generator point. And I'm sure we'll get into, like, what that is. But, like There will be a moment where we we I literally describe what that is using only the numbers 1, 2, 3, 4, and 5 and basic multiplication. So, like, that's how can be learned at this very basic level. I think there are reasons why it's kept from us and nobody it's like, you know, it feels like the hardest thing in the world, and most people are so far away from being motivated to learn it. However, it is simple. An 8 year old, you know I actually have a book. Literally, it sits next to my where I podcast here just to remind me. And it's, like, literally a book about having to teach how to teach this stuff to 8 year olds. And it's this thin.
It's it's it's like, Less than an inch thick. Less than an inch thick. And, yeah, it's it's, his references a guy named Piaget who had a lot to say about gold back in, the 1800 as well. Very interesting. So but the bottom line is, you know, we're all gonna become children again. And that's actually how we're going to, that's how we're gonna navigate through this.
[00:05:22] Unknown:
Yeah. Well, the and the the understanding of math. Right? Because you can even if you learn how to code. Right? So I'm I'm a developer by trade. Like, I get paid to develop. And there there are many such people in the Bitcoin scene that are doing all these all these dev work and stuff. But then you have, like, this other this, like, next level class of Bitcoin citizens, we'll call them, which are, like, the cryptographers and the people doing the implementation, the the application of the cryptography and everything like that. And that's, like, the level, at least personally, that I strive to go to. Right? So I was Yeah. Previously working on, like, FEDIMENT and cashew. And I learned about, like, blinded signatures and, like, the way that equation worked out for, like, how that blinding function works, but the actual underlying math of the big numbers that are used for cryptography, like, the why of it. Right? It's more of, like, the why. Like, why are we using this generator? Why are we using these algorithms? Why does it work the way that it does?
Hopefully, we can kinda dig into it and dissect because that's the next level of understanding. Right? Right now, you as a Bitcoin, you know, if you're listening to your 40 hours of Bitcoin podcast, like you should be, you might understand the macro picture and everything like that. And maybe you understand some basics on how to run a node and how to do some development and everything like that. But that next level, in my opinion, the the the people that I have to trust are cryptographers. Right? So when I get this when I when I'm digging into this codebase, Fedimink codebase or whatever, and it's like, oh, yeah. This is, like, there's, like, this federated, like, consensus mechanism.
Right? And it uses, like, these signatures, these BLS signatures. I don't know what that means. I don't know why that works the way it does. And so when we got to chatting, we both kinda understood from a basic level that, like, there's a next level to understanding this Bitcoin thing. And that's the math behind it. Right? Who do you trust right now? I have to trust the cryptographers. You have to trust the cryptographers. Let's get to a level. Let's try to strive to get to a higher level of understanding
[00:07:22] Unknown:
so we don't have to trust those people as much. Or at least we can at least call b s on something. Absolutely. So let me describe real quick why I ended up in this rabbit hole. Because I am not a mathematician, and I was not I mean, there are people who would call me a financial mathematician as from my career or I was an actuary. But if you're an actuary, you know you're, like, not a mathematician. Right. You know you basically looked at the mountain of math, and then you looked at this job and said, I can handle this job without climbing that mountain. And then, you know, I have a I had a 30 year career where I had to visit the mountain every once in a while in order to do my job. And so I am an experienced self learner when I have to be. Mhmm. So but not you know, I never got a PhD in math or anything like that. But but I did end up in a rabbit hole that now has me look and sound very much the way, let's say, a mathematician in our space, right, asking the kinds of questions and more in in in that process. So let me just let me explain how I ended up here real quick. Yeah. So, like, I am 3 year I am 3 years from my entering Bitcoin, what I call my genesis moment. Alright?
And because of my experience, I guess I came up fairly fast. You know, I under I had a very good understanding of a lot of the sub subjects of bitcoin. I also, like, had a strong disdain for the knowledge system and, you know, so I was consuming bitcoin material. It was very it just felt very normal and it, you know. Yeah. It got I I came up fairly quickly and, one of the first things I found I'll I'll say 3 or 4 months in that captivated me was the base 58 curriculum. And so then I got really into that, you know, coding transactions and wanting to look at the GitHub and, you know, probably within 6 months I'd been to the git Bitcoin GitHub. I had a good grasp of, like, command line transactions.
I went through Jimmy Saum's programming Bitcoin book. And I had a pretty like, you go to the GitHub of Bitcoin, and you see how it works. You really can. Okay? And it's not you don't have to be a mathematician or even a coder to, like it's almost like it's written out in English. You know. So that's really that was really great. And then at some point, I can't remember which book. Like, there are a number of programming books that aren't widely talked about, but but they're really good about programming Bitcoin. I had this this thought that I had to go to, now the sec 256p1.
[00:10:09] Unknown:
P 256k1.
[00:10:11] Unknown:
Yeah. I had to go to that GitHub because now okay. I'm done relying on what the Bitcoiners did. Right? I understand what they did. I under you know, now it's time to understand what, Satoshi was relying on and what we're all relying on for that. And when I went to that GitHub, and this is, you know, like two and a half years ago, almost to the day, I saw I did not see a formula. I didn't see any variables or formulas. All I saw was just an unending sea of hard coded numbers, comma, the limited. And I was like, what the what the fucking hell is this? What is going on? How can anyone understand what the hell is going on here? Right? I thought it was gonna be like a road map, like Bitcoin is, like, you know, like the Bitcoin GitHub.
Yeah. Logic you can follow. That's what code is. Code is just logic that you can follow along. So, like, one of the things I used to be really strong at in my career was, if you gave me an Excel spreadsheet, even if it had 60 tabs in it, I would always be able to find where the the inputs come from and trace, you know, from beginning to end to understand that system. And so I you know, that this is how I that's my formula for understanding things. And the the the sec t 256k1 GitHub was just, like, it crushed me. Like, it literally, like, crushed my soul. It's like I was so I I was on this path. I consumed so much. I understood this bitcoin protocol.
And now I'm at this point of great reliance. Right? And I get nothing. I really get nothing. And there's nobody that can really tell me what to do. Right? There's certainly nobody in my network that I can ask Right. What do I do about this? Right? So here I am. Well, I mean, the answer the answer was, I guess, I need to go buy a cryptography book and start there. And I'm holding up I'm holding up the book I ended up buying, which I thought I think in retrospect was a home run. I'll just tell you. We'll put it in the show nights show notes. It's called Understanding Cryptography by Parim Peltzel is the author.
And one of the reasons I think it was a home run was, one, there's a lot of resources online related to this book. So, like, I'm pretty sure there's a lot of people who have taken taken cracks at all the quest, you know, solutions and you can see all that online. The authors put a lot online. And, it turns out I found out that there's a video seminar by the authors themselves, like, back when, like, open co open open courseware for MIT started. Yeah. I don't it wasn't part of that, but they did this. And, fair warning, if you do go watch them, most of the comments are telling them to get better cameras and that they're getting nauseous. But that's how early, like, they Yeah. This book was written in the late nineties. And so, I loved that it was before Bitcoin.
Right? And that's one of I like to read books that were before Bitcoin, especially on cryptography before Bitcoin, so it's not, you know, it's not super tainted by it and I have to, like, do more rewinding to sort the story out and their motives.
[00:13:40] Unknown:
Well, one of the things I remember on the intro in this book or one of the first chapters, because, I'm into chapter 2 now, but, like, in the first chapter it was talking about, like, sort of the history of cryptography or history of, cryptography. And it goes through, like, the general, like, there's cryptanalysis and all these other, like, different subsections of it. But it largely said that, hey, up until a certain point, cryptographers, like, the the mathematicians behind in doing the application of all this, were very much like almost like a state secret. Like, There was not this, like, open, open source, openly available course material or people out there talking about it. It just wasn't something like, if you were good at math and it led you to this, like, cryptography field, you're most definitely scooped up within the university system and put to work by, you know, some government somewhere. NSA, you'll be yeah. Something something like that. Right?
And only within the past, probably, 2 decades would be would would it has it really sort of sort of opened up. And I found that to be fascinating. And what that tells me too is well, first, let me say this. It's, like, whenever you open source information, which information wants to be free, right? When you when I share an idea with you, now there are 2 ideas that exist in the world, both in both of our brains. Right? So this inflationary nature of ideas and information, allows allows it to develop rapidly. And we see that in our modern world, like, information just proliferation everywhere. But if this idea of doing cryptography and and the math behind it and the techniques and everything like that, if it's only a couple decades old, just like many other things in our modern world, like, we're just starting to see, I think, the impacts of that. And Bitcoin might be one of the first largest impacts, let's say.
Right? I know Bitcoin was sort of the culmination of all these different technologies that came together to to create this this digital money. But what it's done now is it's intrigued and then also unlocked time for people that have the the time and the desire to, like, look into this idea that is cryptography, into this mathematical,
[00:15:48] Unknown:
you know, into these math equations. How does it work? Why does it work? That sort of thing. Yeah. And so something interesting is there's another thing. If you go back 20 years so there's another thing that didn't exist 20 years ago, which is you could not self study math. You didn't have access to even textbooks. So, like, the books that I have in my life, like, they weren't available. And when I told you before, like, there were times I did have to self study math in order to be able to, like, do my job. I would go, I live in Philadelphia area. I would go to the University of Pennsylvania bookstore, pretend I was a student, and go into the back where they had the textbooks for their students.
And that's the pretty much was the only way I was gonna be able to get a textbook to self learn. And MIT OpenCourse, software was really the first, I wanna say 15 year I mean, it was around 08, 09. It was, like, magically the same time that, Bitcoin was launched. You know, we also did have a financial crisis around that time too. And I think a lot of you know, like, whoever Satoshi was wasn't the only one who felt compelled to put something open source into the system for everybody at that moment. Right? Yeah. And, education started becoming much more, available open source and but it was unheard of. Like, you have to know it was absolutely unheard of. If you didn't have access to a university, you were not gonna get these books.
And, you know, you didn't have Amazon. You just didn't have I mean, you have, like, eBay. I don't I don't remember ever thinking I could get a textbook on eBay. So, like, it and so it's just one of those things that also the so cryptography, I mean, we have no idea the story of cryptography. We have no idea, like, what we're screwing with right now and even have trying to have this conversation. We have no idea what kind of secrets are still locked up in, you know, by the people you mentioned. Right? Mhmm. No no idea. But we do know that some we we we know that some of it, enough of it was leaked out. Right?
We're almost like it's like, you know, the MK Ultra experiments led to psychedelics leaking out and people having growth experiences as a result. Right? I mean, God knows why those things ever existed to begin with. But, cryptography, like in yeah, so you talk about the book. You know, they go back to, you know, they go back to Caesar time that, you know, they people love talking about how cryptography goes back to BC. Right? And it's it's an innate human desire to want to protect communication. Right? Yeah. So, this book does a great job of that. This book talks about like, I never heard of DSA I'm sorry, RSA or AES. I'd never heard of any of those.
I knew from my Bitcoin books now what a digital signature meant, what an elliptic curve meant. But, the book the other thing the book did that was really great was went through the math. Now, unfortunately, the math was foreign to me. So, that's what opened up the next rabbit hole. Right? It was like, oh, my God. I gotta now study number theory, abstract algebra. I mean, it felt gettable. It felt understandable. Like, you know, sometimes you look at a math subject and you're like, that's out of reach for the way my brain works. Right? Like, that's never good. I'm never gonna learn that. The the I started to see that the math needed to understand the wall of hard coded numbers was approachable within my lifetime.
[00:19:34] Unknown:
So, like, if I just got started getting to work, I might get some I might get somewhere. Right? It's sort of It's building the context around it. Right? It's like anything else. So you don't know until you can understand the surrounding context. So it's like, if you don't know the terminology, if you don't know the way things are structured, And this is applicable to basically anything. Right? This is one of the things, like, I was I was a linguist in in my past career in the Navy. And it's all about just building context. And so that's why, you know, immediately down the rabbit hole, I started listening to Bitcoin podcasts. I was talking about whatever. And you don't know anything when you start listening to these things. Right? And you hear these words and you're like, I don't I don't know. Like, when they start talking about, like, bond stuff or whatever. Like, I would hear the words and you can maybe understand the basic words.
But the the context of these words together have implications. Right? Like yield curve control. The first time I heard yield curve control on, like, a Bitcoin macro podcast, I was like, I don't know what that means. But I kept listening. And eventually, over time, you start to build the box around, and and you see the shape around these things and see the association. It's almost like you're building up, like, your LLM in your head. Right? Like, LLMs are just relations of of words to each other that occur in specific patterns that can be regurgitated. But human the human brain sort of works the same way where if you don't know the shape of the box that you're looking at or or the the the surrounding things, it's like the force for the trees type of deal. Right? You can stare at this one tree forever, but you kinda have to zoom out, zoom in and out at different times to understand what's going on around it. Like, what's adjacent to it? What's what do these words mean? Language is so imprecise in and of itself.
Context is critical.
[00:21:15] Unknown:
Yeah. And then so then where's the math fall into that? Is that so, like, when you get bombarded with some you learn something from somebody else. Right? Like, you read a book or you listen to a podcast. It takes a really long time to ever actually develop a thought for yourself, asking if this is right or wrong. Right? You're usually just gonna accept that it's right, especially if you read a book. They're like, oh, they wouldn't have printed it in this nice font if this was wrong. Right? You you read a cryptography textbook, you're going along to you're going to assume that it's right. You know, you're gonna and it takes I don't even know what it's almost spiritual at this moment where you start to say, what if this person might what if this is wrong? How do I know this is right?
How do I know I'm right? Right? Mhmm. And this is gonna sound strange, I think, to people who listen to this say this for the first time. But at a very deep level, for me, the learning math is the process of building conviction in your own reasoning. It's not just learning how to implement code. It's not just learning how to validate code. Right? I think most people can know that 10 plus 10 equals 20, and that's not but they don't ask themselves how they know. Like and there is a really deep question there.
[00:22:40] Unknown:
Yeah. Right?
[00:22:42] Unknown:
It's like, the great there's the man known as the greatest mathematician to ever live, his name is Gauss, Carl Friedrich Gauss. And in the I read his biography recently, one of the biographies, and it talks about how when he was 3 or 4 years old, he was adding large numbers. And there were no calculators to validate any of it. And the adults just thought it was cute. But if you ask him throughout his life, he always says he knew he was right. And, you know, you have to ask the question how. How does one know they're right? And when it comes to people who are relying on Bitcoin for their life savings and more, right, so, like, the longer you're in it, the more you're like, it's not just about my life savings, this is about my livelihood, Right? My livelihood is based on not caring, not depending on anybody for any aspect of my sovereignty, really. Right?
It's really important that you build a sense of conviction in your own self. And there's a lot of ways to do that. So you can, engage in self defense. Right? You can work out. You when you work out, you have an experience of yourself being strong. Right? And then that builds conviction, and then you work out more and better, and it becomes a cornerstone that you build on.
[00:24:09] Unknown:
Well, it exposes your weaknesses too. Like, when you go to work out and you hit something that you can't do.
[00:24:15] Unknown:
Yeah. And that's how you become strong. You break that muscle and rebuild it, and you give it the stimulus. You send it messages that you're not strong enough for this. So, you know, you're gonna have to get strong enough. So learning math and it's not the only way to build conviction in your reasoning. You you can study logic. You can do things. But, you know, it's really low hanging fruit to just get on, spend 15 minutes or 20 minutes a day learning math and filling in that part of your fitness. Because getting we this thing is so early. Right? This Bitcoin thing is very early. We went through a fork war in, you know, 2017, which I would argue is still is, you know, never ended.
But it's decided, but not certainly not settled. Right? That man, I think I'm really lucky that it was one of the first books I read when I got into the space. Because it's like the one book that I think has the most to do with my conviction is that one because it's the one that made me think what, really, what are the attack vectors? And when I started thinking about the math thing, so first, you know, yeah, I run into this GitHub, and it gives me nothing. So how do you so we have a we have now a society of Bitcoiners that can't validate cryptography. That's was my first thought. Right? Mhmm.
That was so when I say that I fell apart and I had a crisis I had, like, a personal crisis, it was like, oh my god. Who is minding the store here? Right? And then so is it up to me to start in this rabbit hole and try to answer this question? Because, otherwise, there are if there are people that do understand this, they're not sharing it publicly. They're not writing books to connect this dot to us. Right? The people writing books are connecting, you know, of the world are writing books on what they know. The people that can answer the question, what is that wall of numbers on the on the secp GitHub, I don't know that they're intentionally keeping secret. They may not think it's interesting to people, but they are not writing books about it. And it is a, to me, a glaring weakness in, our ability to be resilient to, you know, I'll say to a social attack.
And make no mistake, like, I started, the more I started getting exposed to, like, mathematicians and PhDs, I started, you know, hundreds of mathematicians whose YouTube sites I use, and then I go check out their biography. They all work for shit coins. It's like, wow. Fucking weird. Right? Yeah. That's weird that they all are, like they all want to do blockchain research and stuff like that. And then it's, like, no. It's not weird. That makes total sense. They're the that's who pays. Right? Yep. And then, you know, so then there's this small group of people that aren't doing that, like the Peter, you know, Peter Willis of the world. And but they all work for 1 company for the most part. Yeah.
And we don't have now, like Andrew Polstra, I've met him and I've found him to be very candid when I'm and ask answering questions when I meet him, but it's not like I can't ask him the question I had.
[00:27:40] Unknown:
You know? It's like you said earlier, like, people in your network like, you can bring these questions, but there's not anybody in your network to to tap into to get them answered or to know or to trust or to understand from.
[00:27:53] Unknown:
And that's Yeah. So I have this fear, like, what if, you know, what if one day the the guys we trust wanna set up a fork and they're they're convincing us that everything is cool, trust us. Maybe they can be trusted, but who could, you know, who could validate it?
[00:28:09] Unknown:
Right.
[00:28:10] Unknown:
Right? Who could validate it and tell everybody either it's cool or it's not? And that was what scared me. Right? Because I remember I read that book, Block Size Ward, and I was like, wow, people were pretty sure of themselves. It really impressed me. It was like, wow, people really must have understood Bitcoin really well to have understood that increasing that block size could have such an impact. But that was a non mathematic you know, they weren't using mathematical weakness against people. They were just using you know, this was like It's an economic thing. I would just say the attack was early and Yeah. Like lame. You know? And it should've worked anyway, and it didn't because peep you know, but, you know, it's a beautiful thing.
But to me, the attack the vulnerability of mostly, say, Bitcoiners who don't give a shit, like, don't give a shit about math
[00:29:04] Unknown:
is very much out there. Right? So where do we begin giving a shit? Right? Like, where what what where do we start?
[00:29:11] Unknown:
Where so I so I always I have 3 kinda key reasons. So I basically said, man, I think I need to start a math institute somehow. Right? We somehow, they're and somebody needs to put energy into making sure we get better at this. Yep. And I had 3, like like, 3 key things I thought would come out of there. 1 would was, like, personal sovereignty. Like, you can't have you can't be relying on somebody else to do your math. Right? It's like, okay. You're really not sovereign if someone's calculate you know, if someone's doing calculations for you. It's you know, my grandfather was accountant for people, and I just remember how much people relied on him. And I remember thinking, man, that must suck for those people to have to rely on somebody for their own business to know what's going on. Right? You can't be personally sovereign if you're asking, you know, understand there's too many incentives to to, you know, for that to go badly. And the other so, the second thing I thought about were homeschoolers and the rising tide of homeschoolers that have no conviction in their ability to teach their kids math. And, like, what are they gonna do when their kids start showing a lot of talent Yep. And it freaks them out. And then, like, are they gonna break and send them to a quote unquote real school because they don't feel like they can do the job adequately at home? Or they they lack they don't know where to go. Right? They lack the resources. Something like what we're trying to build doesn't exist.
And they don't know where to go to even, you know, to even know what to do. Right? So I saw that as a bit. And then and then I saw the just the fork possibility and just, like, like, god. Everyone just needs to level up, like, a little bit, and we will be a lot stronger.
[00:30:56] Unknown:
Yeah. Well, one of the recent, you know, topics du jour in the Bitcoin ecosystem, which I don't think was gonna go away, it's sort of an evergreen topic, is this whole idea of, like, quantum resistant things. Right? And and if so somebody is proposing some new signature schema. For instance, I think last year, sometime last year, the NIST, NIST, it's like the National Institute of Standards and Technology or something, released these, like, candidates or selected candidates or something for post quantum secure digital signature algorithms. Right? Yeah. So the post quantum I went to go look at them. Yes. This is quantum attack. When I went to go look at them, but, like, I don't I don't know how to, like, how do you know, I could probably go through the meeting notes or something like that or some resource of, like, how they determine these are the ones.
But without understanding math, the math fundamentals, the math basics Yeah. Of how all this shit works,
[00:31:50] Unknown:
I I'm I'm looking at Greek. Right? I'm reading Greek and and it doesn't make sense. Know what the problem you're trying to solve is. So I can tell you, when I read this book, Understanding Cryptography, I don't know if it was chapter 1 or chapter 2. It was very early on, I became very comfortable that at least with the fact that it was not a difficult exercise to raise the hashing out, you know, to raise the hashing out, though. If you wanna go from shot 256 to shot 512, it's not it's like a pain in the ass. It's not it's not an, you know, it's not a game over. It's a pain in the ass. Right. And, I became comfortable. So I think, you know, like and I think learning in a lot of ways, we're dealing with time machines here. K? So, like, a hashing algorithm is like an anti time machine. Like, it will take you literally forever to the end of time unless you know the one thing you need to know to
[00:32:43] Unknown:
Find the solution. To undo it. You would just will never, you know, it'll never happen. You can't you can't muster enough power. That's why I say These are called one way functions. Right? Like, hashing algorithms that were or generally referred to as a one way function. It's like a smoothie. You put bananas, strawberries, yogurt into a blender, and out the other side, you get a strawberry banana smoothie. Right? But you can't take that smoothie and go back to the exact banana, the exact strawberries, the exact amount of yogurt. Right? You just can't you can't do that. That's right. So Hashing Algae is
[00:33:14] Unknown:
largely that same way. Now there's a a very simple mathematical concept, which is, you know, you just look at both directions and what it's gonna require. You just can you can see what calculations, what bit operations are required to go one way versus the other way. And that's, you know, that's something I think we wanna demystify. And I so there's a certain level of things that just need demystifying. That will make a big difference. Then there's this level of things that are, like, they're kinda difficult, but they're so cool that they're worth learning. And then there's things that are difficult to the point where we're we are just trying to touch it. We don't know it yet. So I would I would classify most of, like, what we wanna cover in those three categories. Most of it, I think, is demystifying.
Okay. And once it's demystified because that's the thing. Like, what gets weaponized against you is, like, your infinite sense of what this could be. And you you're you're in this Dunning Kruger. You know the Dunning Kruger effect. Right? Where, like, people who don't know what the hell they're talking about can sound really smart because they're not stopped by their even their own incompetence. Right? And so they talk very confidently. And people who are competent see them as, oh, shit. Maybe they actually are onto something. And I don't know anything about this. So this is an infinite world of trouble for me. And we wanna take that infinite world of trouble, turn it into a finite world of trouble, and then and then make it smaller and smaller and smaller to the point where it's it's like, well, no. This is something I can handle. It's like, you know, imagine you your personal finances, you never did anything before. You never touched them. You never looked at it. And now, all of a sudden, you have an event in your family or something, and you have to now handle it. It feels infinite. You're like, oh my god. This is just an infinite thing. And then you crack you you crack it every day.
A little bit Clarifying the context. Right? A little bit every day. Yeah. And then it becomes a smaller and smaller world to the point where I mean, this is probably if if anyone here knows, like, understands cryptography, that's definitely how that worked. Like, when you first started, it seemed infinite. Right? Yeah. So we wanna do that with the math side because that's the thing that probably affects the most amount of people. There's probably more people in the world that just are intimidated unnecessarily to the point where they lose power. They cede power over it. Right? Right. So that's what we wanna, like, demystify a lot of this so that you no longer cede power over not knowing it. And then you can start actually understanding and build build a conviction for yourself.
So so what's the first thing? You'd mentioned a few before we started recording. What what, like, what was that first thing on your list? It was a list of things that we were gonna go over today as, like, the entry point. So there's a thing there are the things today, which is, like, the the the, you know, what most books would call section 0. But, like, when I talked about a general curriculum, I think when I was I was saying general number theory, we're talking division algorithm, Euclidean algorithm, Chinese remainder theorem, and, you know, then like finite fields. And these all, like, it's so weird. Like, my daughter is a math major. And she would be intimidated by those words for the most part, you know, because they don't you don't even learn stuff you don't learn that stuff until late.
You know, math majors learn calculus and how they can apply it to physics for the most part, for, like, the majority of their math majorhood. And then if they survive all that, they get to learn the kind of cool stuff that, you know. So that may sound that may sound intimidating. Right? But, like, the division algorithm is basically just how you divide numbers. And, it's like, if you remember when you were in 3rd grade, you did long division and you basically said, okay, I have something I have a factor that does multiply this number and then I have a remainder. And the algorithm is basically through a repeated process of finding remainders, you end up solving this division.
And you get into this thing called greatest common divisor and find that numbers that are relatively prime to each other, meaning that they have no greatest common divisor. These are important concepts in cryptography to understand, eventually understanding public key cryptography and very basic, like, discrete log type concepts.
[00:37:57] Unknown:
Yeah. Because when you divide it, it was like, I remember the the little drawing. Right? It was almost like, like a curved vertical line and then a straight line. And you're dividing the thing on the right side underneath this little drawing by the thing on the left. Yep. And if the thing on the right was, like, 100 and the thing on left was 3, it's, like, okay. Let's figure out, like, how many times 3 goes into a100. And then you write the answer on the top, and maybe there's a remainder. Right? That's right. Sometimes there's a remainder of 0.
[00:38:27] Unknown:
Those are nice. Right? We consider those nice. Right? Right. And then when there's a remainder that's non zero, we're like, ugh.
[00:38:34] Unknown:
But that's critical for cryptography, though. Yes. Because that remainder math
[00:38:39] Unknown:
is one of the things that I think we'll get into, in fact. Yeah. One of the things I was thinking about for section 0, we didn't even it was not even on my list, was just talking about the different number systems that we know about and exist, like the integers, the rationals, the reals, because there's a system of remainders, Right? That which is what we end up in a with in a prima in a finite field, called residues. System of residues modulo a prime. And, which, if anybody has opened, like, Programming Bitcoin by Jimmy Song and did that first three pages Yep. You you have a prime number like 5, and then so this this number system of the residues, modulo that 5, would just be 1, 2, 3, 4.
[00:39:29] Unknown:
And It's 0.
[00:39:31] Unknown:
Well, right. But then when you, and 0 is implied. Yeah. It's called an equivalence class. But, like, that's a number system just like all the integers. Right? But that's a number system that possesses the very special properties of like closure. Meaning, like, if you add any two numbers in that system, 2+4 is 6, but 6 modulo 5 is 1 and that's in your system. That's in your 1, 2, 3, 4. So, like, you can't do that with the integers. You can't do that with the rationals or the real numbers. You can only do that in this system of remainders.
[00:40:06] Unknown:
Right. So naming I guess, naming these, like, groups of numbers and stuff like that, you can you can, again, contextualize. You can put a box around what this is. So if we're talking about, what are they called? They're called sets or fields? Or
[00:40:19] Unknown:
So fields is a very specific structure. Mhmm. Right? And a finite field is an even more specific structure. It's a field that has a finite amount of elements. This happens to be the thing we care about in the cryptography we use around Bitcoin.
[00:40:36] Unknown:
So in your example, though, of this this finite field, I think it's referred to of the order of 5. Right? Yes. Yes. So finite field of the order of 5 is 01234.
[00:40:51] Unknown:
Yeah. But we don't we but those actually, the 0 is not the like, the 0 doesn't go in the field. It's not it's it's it's, it's not part of that finite field necessarily, because it's already it's already, like, it's already equivalent. The remainder of 0s is already equivalent. Not to get too not to get, you know, not to get caught up in a, you know, in a detail. What matters is that if you multiply any two numbers in those fields, you get a number in that field. If you add any two numbers in that field, you get a number in that field. And, yes, a 0 could be, you know, 0 could be a 0 could be one of those numbers.
Right. Right? You could add 4 and 1, and you get 0 because it's the remain remember, we're talking remain the the the members of this field are remainders after dividing by 5. Right?
[00:41:46] Unknown:
So, like whatever the operation is. So if you're doing 4 plus 1, no matter what you do, you do that normally. Right? So 4+1 would be 5, but 5 is not within it. We've exceeded the bounds of this finite field. And just no matter what you do, you always do if you're if you're exceeding the bounds of the finite field, you do a modulo
[00:42:07] Unknown:
Yeah.
[00:42:08] Unknown:
Or a division, and then keep the remainder. The modulo just means divide by this number, and then whatever the remainder is is the answer.
[00:42:17] Unknown:
Yes. So, like, it's it's another number system, like the integers. Right? It's that's and I don't think like, for me, I spent 30 years, like, learning a lot of math, and I got to this thing about finite fields, and I almost shut the book after 2 pages because I just, like, couldn't, like, I couldn't rock it. I was, like, what the heck is like, there's no there's no system where if I multiply two numbers, it is in the field. Like, that's crazy.
[00:42:48] Unknown:
Well, clock math. This is clock math. Yes. I think colloquial, we can refer to as clock math. Yes. If you have if I've had 2 o'clock, 2 PM, 1400 well, we'll use 1400 for military time. Yes. And you add 56 hours to it. Yes. Right? You don't add 14 plus 56, which would be, like, 70. It's not 70 o'clock. You have to once you reach the end of that finite field of numbers, which is 24 hours in a day, you sort of loop back to the beginning again. Yes. Great example. You keep doing that over and over to stay within the bounds of it. But the quickest way to do that would be, you know, 14 +56modulo24 is sort of the rep way you could represent it, which means 14 +56, which is 70 divided by 24.
And then whatever the remainder is is what you're at. Yep. Is is the answer. Which I think is 22. Well, I was just about to punch it in. I don't know.
[00:43:50] Unknown:
And I yeah. I got that because 7 it I know 72 is a multiple of 24. So negative 2 +24 would be 22. Let's find out. I don't know. Let's find out. Big test on fundamentals, mental math here. Please be right.
[00:44:11] Unknown:
56 hours. Hold on. Alright. Let's say say it out loud again.
[00:44:15] Unknown:
You had 4 you were at 14 o'clock. 14 o'clock. You had a 56 hours. Yep. Right. So you got to 70 o'clock. But there's no such thing as 70 o'clock. We only go up to 24.
[00:44:25] Unknown:
Right? So we divide by 24.
[00:44:27] Unknown:
And you probably get a remainder of 2. So you got right? You got 48. You can account for 48 of those hours of those 70 hours. Right? And then what's left is 22 is what's left is 22.
[00:44:41] Unknown:
Yeah. So 70 minuteus 48
[00:44:43] Unknown:
is 22. Yep. That's that's your that's your new remainder. Right? Mod in a modulo 24
[00:44:51] Unknown:
work. So 2 PM, 1400 plus 56 hours is is 10 PM. 10 PM. Yep.
[00:45:00] Unknown:
Cool.
[00:45:02] Unknown:
And that same principle is how these elliptic curves work. Correct? Very similar. Yes. And,
[00:45:10] Unknown:
probably there's a good I mean, at some point we can get into why visually this makes sense. But, like, if you graphed a ellipse, right? You graphed an elliptic function in real like with the real numbers, you would see a drawing that looks like an oval. So, it's pretty easy to look at the oval and find what points made what points gave you that oval. Very easy to do with the real numbers. Right? You see the ellipse, that oval shape, and you know the formula. Right? It's something like x squared over a plus y squared over b equals something. And you're like, okay, I know it's an ellipse centered around, a point. Right? And I can figure out exactly where the input is.
But when you locate it in a clock math type field, it becomes a scattered thing that you can't locate the origin of. That's why it's used. And that is why these things are used in cryptography. That, like, and in the in the book Programming Bitcoin and in the I'm pretty sure in the book Mastering Bitcoin by Antonopoulos Mhmm. There's some really great visual examples of why,
[00:46:30] Unknown:
you know, of how that works. Right? And and the equation that we specifically are talking about, this this secp256k1equation, is y squared is equal to x cubed plus 7. Yeah. That is, like, that is the definitive question. So if you go on, you know, like, some Wolfram Alpha, you know, Matlievs type of site that you can plot these equations, If you just punch that equation into there, it will plot it for you, and it'll it'll it'll give you a nice illustration.
[00:47:01] Unknown:
And you could but you can be in 7th grade. You could see a picture of that elliptic function, really. Right? Mhmm. And you could say to yourself, yeah. I have a pretty good idea of what the x I I have a pretty good idea of what the x and y are. Right? Yeah. I can kinda visualize it pretty well, but when you plot that and this is a this is going to be hard to explain and visualize. It's just if when you plot we'll we'll put a picture of this in the show notes from that book. But when you plot it over this numbers not a real number system, but a finite field number system, Modulo a certain prime, which you can go on bit get Bitcoin's GitHub and see what that prime is. It's gigantic.
Right? But if you, in the book, they'll they do it and they'll say, oh, let's, if the prime was like 31, something very manageable, you can see. It's, it looks like a, it looks like a random scatter plot is the bottom line. It's when you plot it over a finite field, it looks like a random scatterplot. So when you look at a point on that field, you have no clue where it came from.
[00:48:11] Unknown:
So it's almost like these these numbers, these these groupings of numbers. Right? Like, real numbers. I don't know which one it represents. Yeah. When we're using real numbers, these equations take a a recognizable shape. So they're they're predictable in a sense because when we plot them out, there is a shape, and and, you know, humans are great at shape, pattern recognition, and everything like that. And so if you apply these finite fields using these prime numbers to this shape, it completely obliterates the shape to the visual eye. There are still patterns. Like, it's still following a set equation.
But the it it it's almost like it's, it's like obfuscating what the true shape of this group of numbers is, even though there is a pattern to it. Yeah. So it's a bit it's it's a visual
[00:49:00] Unknown:
explanation of why these things are why it's cryptographically, you know, why is it used in this way? Right? How how is this used to obfuscate the origin, right, the origin of the point? Like, at some point, we can we'll get talk about public key cryptography. The whole point is I can tell you a point on the curve, right, without telling you where the origin was. Right? That's the whole kinda that's the big premise. It's the beauty of it. Right? I can I can tell a partner I wanna work with what my, what the answer is? And then if he can guess how the answer was created, then he can, you know, he can move something.
Right? He can access this he can access something that I have to offer him. Right? It at some point, you have to be able to put out there something in public. Right? Yeah. That can't be backed into. And so this concept and this is why, like, we talked just section 0 a little bit about the number systems. Right? Because you it's if you understand, a little bit about there are these number systems. 1 is, like, the integers, very simple, like, we call whole numbers, like 1, 2, 3, 4, also negative 1, negative 2, negative 3. Right? Those it's just those. Right? And then you have this other number system called rationals, which is like, these are now, you can make fractions of integers.
What we call P over Q, where P and Q are integers. So you have 1 half, 1 eighth, onesixteenth. Right? 3 28ths. Those are now rational systems. So you have a little bit more number. You have more numbers than in the integer system. Right? And with more usefulness. For example, the integers, you can't take it back, you can't invert an integer that's not 1. You can't, in other words, you can't multiply an integer by another integer that gives you what's called the identity, I. E. 1. You need a rational number for that. So if I had the number 3 and I wanted to get to the number 1 by multiplication, I need 1 third.
So we created the rational numbers to be able to do that. Right? And then, there's, between the rational, from the rational numbers, you have a big leap to what's called the real numbers, which is, now, that includes numbers that can't be represented as a fraction, like the square root of 2, like pi, things of those natures. And there are a lot of those numbers. That's a, I mean, it is a field in and of itself understanding the real numbers. But with the real numbers, we get to see things like continuity. We get to see continuity between the numbers, which we never really could see in rational numbers.
[00:51:59] Unknown:
What do you mean by continuity?
[00:52:01] Unknown:
In other words, do you ever draw a function, like a function, and you can see how the line doesn't break as you move across the x axis. The numbers the numbers represented on the y axis by the function do not break. Whereas, if you all if all you had were integers, you would have dots. If you had rational numbers, you'd have more dots, a lot more dots, but you'd have a lot of gaps. In the real numbers, basically, we say the rest is are the gaps. So that we can now have things like continuity, which gives us calculus, you know, gives us optimization, gives us all derivatives and integrals, setting derivatives equal to 0 and finding maximums and minimums. Like all of that is because of continuity and we get to do that because of the real number system. That we get that to because we get to imagine that it's complete.
Right? We get to imagine that there's absolute completion. Right? So that system was created really to fulfill that. You know? Somehow, we had this innate ability to think about it, but without the number system to work with. Right?
[00:53:14] Unknown:
But in numbers, we we we use numbers to, like, sort of try to represent the real world. Right? Like, the this this idea of, like, what is this table? Right? There's there's you could break this table down into the mathematical formulas that constitute, like, the atoms that create it and everything like that. Like, I feel like the more we try to explain the world with numbers
[00:53:37] Unknown:
Mhmm.
[00:53:38] Unknown:
And having these different sets of numbers or or, to to use is a way for explaining different aspects of the world. When you're talking about the the the real numbers and, like, pi being one of them, my mind goes to, like, oh, that's why pi is part of, like, a circle. Right? Like, how do you represent the idea of a complete circle?
[00:53:57] Unknown:
Right? Yeah. Like and so and it's funny because if you started with, well, I thought let's say I started with a square or triangle. Right? Or square or Mhmm. A poly you know, and I I say, okay. Well, I have a square, and I see how a square is kinda like a circle. Right? If I didn't have a circle, I'd probably just say, oh, well, then just give me a square. Right? Right. And then because I can make a square at 4 sides, 90 degree angles. Right? Now, okay, what if I had a pentagon or a hexagon or a decagon? You see, the more sides I add The closer you get to a circle. The closer I get to the circle. But you have to imagine this idea that I'm gonna have infinitely many sides in order to actually have a circle.
[00:54:45] Unknown:
Yeah. Yeah. It's the it's like the granularity. You see that sometimes, in, like, image displays. Right? Like the the poly polygons, the number of polygons that go into an image. This is something like video games and stuff. It's like the number of polygons. Like, PS PlayStation 1 had, like, low polygon resolution. Right? It's why, like, Lara Croft looked like a box when she was running around. Right? But in the modern days Money for Nothing video. What's
[00:55:11] Unknown:
that? The there's a video for a song called Money for Nothing.
[00:55:14] Unknown:
Yeah.
[00:55:15] Unknown:
That's I highly recommend everybody go check that out on YouTube. It's, iconic song from the eighties by Dire Straits. And, they really used at the time cutting edge CGI. But if you look at it now, it's it looks like the most primitive thing ever.
[00:55:34] Unknown:
Yeah. So, I guess, it's it's we're further we're getting further resolution in in in by using these different numbers. And so where do where do where do finite fields fit in? Is that its own It's another number system. So it's another type of number system
[00:55:50] Unknown:
to answer a different question. So, like, the real numbers were You might say the real numbers were created to answer how do I solve a question called, x squared minus 2 equals 0? That's know, at which which, Pathagoras asked. Right? Mhmm. That's a question from the days of Pathagoras. And it's a there's a really interesting story that, you know, so, Pythagoras was the king of the right triangle. Right? Mhmm. And the theorems the theorem his theorem, the a squared plus b squared equals c squared, which were the two sides of a right triangle, if you squared them, they became the square of the, what's called the hypotenuse? The hypotenuse. Yeah. The long side. Well, if you have, if your legs are 1 unit, which people love to think about anything using a unit of 1. So what if you had your sides 1 unit?
Right? How do you represent the long edge of that triangle? And, you know, so Pythagoras was a he was a bit of an influencer in his time. Okay. And he had a group there were a group of people called the Pythagoreans who followed every all those teachings. And, there was we they knew what rational numbers were. They didn't really know what irrational numbers were. They just knew that they they knew there were numbers that couldn't be represented by well, at least they thought there had to be, possibly, numbers that couldn't be represented by fractions. Right? Mhmm. But, Pythagoras completely rejected this idea.
Pythagoras thought absolutely the existence of an irrational number is absurd. And, one of his students basically said, yeah. But what if your what if one of your triangles, you know, has a long edge square root of 2? That's, you know, square root of 2 is irrational. And you know what happened to that guy? His name was like Hep Hepha Hepha sis. Hepha sis. You know what happened to him? A bunch of the Pythagoreans took him on a boat and dumped him in the ocean, killed him. They didn't they actually killed him And, because this was blasphemy.
Yeah. And they didn't well, it wasn't I mean, and, you know, part of it was like if the world figures out there are irrational numbers and this whole Pythagoras thing is bunk. And guess what? The one of the most, like, popular proofs there is in the world, and people love coming up with different ways to do it, is to prove that square root of 2 is irrational. And it's almost like it's it's a fuck you to the the idea of the Pythagoreans. Right? Right. Slay your this guy's slayed his hero. And, you know, so we, you know, so, you know, we standing at the base of cryptography, we're not that different from, like, the Pythagoreans because we're we're staring into a space we know very little about. We're actually asking, do these things exist? Right?
So they were asking, does the is there even a number system that includes these numbers? And, you know, the answer was yes. And it was controversial. It was so contra you know, it was so controversial. It. And so now the question is, is there a number system that allow that really facilitates, what we know as cryptography and the ability to protect value in a secret. Right? And the answer is yes. And it's called the it's it has a name in English called the finite field, and I feel like that cheapens it. I think that the thing to the thing of the people to get is that there is a number system, right, that's similar to the number systems we know of, but not exactly like it has its own structure and rules, but there is a number system that makes it possible.
[00:59:59] Unknown:
Man, I think we I think we cut it there. I don't know. That's a that's a beautiful introduction to, I think, what we're gonna explore here. Cool. Yeah. And, for anybody listening in, thank you, 1st and foremost, for giving us one of your most scarce resources. And I look forward to going down this rabbit hole and looking at all the cave drawings that you've painted, Fundamentals. Nice.
Introduction to the Podcast and Hosts
The Importance of Math in Bitcoin
Understanding Bitcoin and Math
The Journey into Cryptography
The Role of Cryptographers
The Genesis Moment and Learning Path
History and Accessibility of Cryptography
The Math Behind Cryptography
The Importance of Math in Personal Sovereignty
Quantum Resistance and Cryptography
Demystifying Math and Cryptography
Number Systems and Cryptography
Elliptic Curves and Cryptography
The Evolution of Number Systems
Conclusion and Future Exploration
