Hash Functions, Linear Algebra, and Alice in Wonderland

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Created: 9/28/2026 1:43:57 AM
Duration: 3023.908
Channels: 1
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Hello and welcome
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to the Magic Internet Math podcast.
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What's up? I am Brian.
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What's up? I'm Rob.
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We got Hello, my clinkers. Gotta get the Internet right away for the context loading for the clinker.
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Yeah.
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Well, this is gonna be for some reason we do have tech we did some technical things going on today. So if you're listening to this and it doesn't sound quite the same.
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I'm also on the road.
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So my podcast setup.
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That's right. Rob's a traveling man today for
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some reason at home but not
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able to
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access my equipment for some reason because my computer can't find it. Anyway,
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so
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it is what it is and will be this will be a short one,
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but we wanted to get a rep out to you. It's been two weeks.
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How you been?
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Doing well. Keeping busy.
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Yeah. It's
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you know? It's
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you know, just always
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I was talking to someone recently,
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and
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they said, if you want something done, talk to a busy person.
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It's kind of like a Peter's
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principle of like,
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those who have will get more and those who don't have anything will get less kind of like just compounding.
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And the busier you are, the more things they can put on your plate and the busier you get. It's like this reflexive loop.
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Yeah, it's like an inertia thing.
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Yeah,
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it's easier to add onto somebody who's already running
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than it is to get someone to start running. It's like that Mitch Hedberg joke.
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Like the only thing
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harder than stop smoking is to start flossing.
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What a legend. What a legend. Yeah, like it's so much harder to get somebody who's not like an arrested state to leave that rested state. Right? Totally.
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And Yeah. Know.
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So that's you, dude. I mean, yeah, I guess it's like everyone that is you. Everybody is looking at you as the busy guy to get things done.
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Yep. Nothing you can do about it. Nope. Nope. Nope. Little
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preview. I I also have to do
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I've now gone from, like,
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token panel guy to now I'm getting invited to do keynotes.
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So I'm gonna do keynote at Lugano.
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I am gonna be speaking at Amsterdam. That's confirmed now. Okay. So want both to do our panel,
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but I have to talk to them about whatever else they want me to do. But I'm doing a whole
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excuse me. You guys Yeah.
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But you've heard it here. You guys heard it here first.
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Rob and I now Excuse
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We've heard it before. Even before BTC Inc. Rob and I, we will be doing
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will be doing a magic Internet math panel
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at the Amsterdam conference.
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Oh, yeah. And I just booked my flight last night and it was brutal,
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but now I'm excited.
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Good stuff. Yeah. Yeah.
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And I at Lugano, I'm gonna be doing a keynote.
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You may hear this before they announce it. They before they announce it,
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but the panel's gonna be called risk of ruin. I'm sorry. Not the panel, the keynote talk. So I have to do twenty five minutes on risk of ruin.
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We're gonna be doing we're gonna be getting in it.
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This is good, man. This is good.
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Yes. Things are moving.
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Yes.
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Just like to think about the fact that there
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there's a panel now
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called Risk of Ruin.
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Right? It's like these things are getting on the agendas at conferences. It's be a keynote. It's gonna be a twenty five minute meeting. Yeah. Sorry. Missed the keynote.
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It's kinda like Figure out how to do that now. Yeah.
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But like we just went through so like,
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out of nowhere out of nowhere in the universe in the ether, I somehow made happen the fact that at Prague, there was there were two, you know, a keynote at Freedom Tech and at Prague on
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our podcast essentially. Right? Mathematical layer of Bitcoin sovereignty and now, know, now you're you're pulling risk of ruin.
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And
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I think we're just building on these blocks.
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That's right.
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You know? And it does seem like the world is starting to care
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about, like just the world is starting to care about this stuff. Everywhere I go, people
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wanna talk to me about this type of stuff now. So like I just went to the Midwest Bitcoin conference. I'm sure you got I'm sure when you were at BPI in DC
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Yes. People people wanna talk to you about the math stuff.
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Yep.
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Yeah. It's it's
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the tides are changing.
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People care now.
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Yeah. And it's not like what I'm getting is not a shallow
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interest in the math. I'm getting like a genuine
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interest. Almost like people are worried.
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And they're like worried that they don't know that that they got this gap.
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Yeah. Which is exactly
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where we want people.
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Since our last podcast,
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my episode on Help Money podcast came out
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and with Casey and Aaron. And That was awesome, by the way. I enjoyed that. You. The there was a good part of it where I had multiple people call me in the post fallout of all the cold card stuff asking if, like, was Bitcoin just vibe coded slop? Was like is it like was it all going to fail?
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And I was like, no.
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Because the math works. Like, the math like, you have to put you but if you've never gone to the basement and, like, looked at all the the piping,
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you wouldn't know.
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You wouldn't know.
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Yeah, you got to look at all those ugly electrical panels.
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You know, it's like, mean, when you go look for a house, I don't know how many times you've looked for a house or how many houses you've looked at buying but like you go at least like in Pennsylvania where there's a lot of old houses, you'll go into some of these basements
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and you'll see some franken electricity
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panels just like with wires crossed and just it looks like,
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like it looks like Phil Spector's hair.
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If that if that resonates
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with anybody,
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go Google it. Alright. Go Google Phil Spector. I'm looking. Oh, yeah. It's like a bunch of wire. With it. Yeah. It's mad at dots. Yeah.
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It looks like like in a in a book of when when if you ever open crack open a book of Monte Carlo simulations and techniques, it looks like the simulated Monte Carlo.
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Well, that's actually just a seed phrase that it just branches out to all the addresses. That's that's exactly what it is.
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And it's like, I feel like the gift that we've had is the opportunity to go into this one basement in this one house
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and take a deep, deep, deep look because we care about this one house a lot, right? Like
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our fucking livelihoods
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are,
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you know, we realized a little late after the fact that our livelihoods are dependent on
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everything working here. Right? And this is sort of the kind of thing people are starting to ask you about, like, does this thing really work? And I feel like we went in and we went in there, understood the math. And then after the call card thing, you went in there and red teamed the repo. The code. The code. Yeah. And
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it's starting to feel good. Feeling
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good about it. But that's actually there will be a
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portion of my keynote that's just going be talking about lip sec pee. Like, we're just gonna, like, I'm just gonna make everyone eat their vegetables. Like, we're just gonna talk about this stuff.
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It's funny because I did that in my keynote. I was, it was like, this was pre cold card, right? But obviously, it's a function of you and I talking a lot, right?
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And there was I had a slide that had I called the the three branches of government, right? Bitcoin Core,
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the industry,
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and LibSecP.
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Right.
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And like,
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you know, like if you're in the industry, you might be able to code something, you don't have to, you may not have to live and die by Bitcoin Corp giving you
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what you need, the substrate For you need,
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example, like you figured out
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that miniscript was already there and nobody was building on it and there was a you know, it was like a finding a treasure
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that was already But
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I was saying don't sleep on LipSecP
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being
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like don't overvalue,
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don't overvalue Bitcoin Core. Don't like, and this was in June.
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That was, I was speaking I feel like a little bit into
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the BIP 110 thing where people I think were just overvaluing
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maybe their fears of Bitcoin Core and crashing out on Bitcoin. It's like, dude, you got bigger things to crash out about if they're fucked.
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And
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Bitcoin Core is important. But as I always say, the distance between us
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and Bitcoin Core is like nothing compared to the distance between us and the guys that wrote LipSecP.
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That's why we do this podcast and not a, hey, let's understand Bitcoin Core podcast.
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Like, we feel like we need to start
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reducing this distance,
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even if it's like immeasurable,
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it looks like it's a millimeter
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a day. It looks like it's so slow, it's like the earth turning, but every once in a while you wake up and you see snow outside and you're like, Oh, we did move. This thing did move.
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To me that's,
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if you want to avoid a crisis of confidence in Bitcoin, like one day,
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of all the things that are gonna like really get
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You know, this, what we're talking about here really to me is a massive piece. Just a massive piece of that. Just inch yourself as close as you can
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towards getting a little and we might have a fighting chance
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when the quantum FUD really hits something that we clearly just spent time understanding.
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Yes.
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Yes.
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I totally agree.
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You just gotta
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you know, you you don't wanna have your lack of knowledge shake you out of your positions. Right? So you just wanna you have to just do the work.
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And
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the nature of this kind of the work is like, this is just foundational. This is Bitcoin.
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And it's not fun. It's not a macro talk. We just talk about the government printing money. Like, you So just look at good. Modular arithmetic.
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Do the math, buddy. Just do the math. Yeah. GDP Rate. Debt.
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Right. It's gonna happen. Yeah.
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It is important. It's it is the Sure. The why you buy Bitcoin, but it's not the how.
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Right? And so you need both of those things.
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Yeah. So like it's
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It's not, you know what? It's not money printing that's gonna deliver the value of Bitcoin. It's the backbone we're looking. It's this it's the cryptographic backbone. That's Why what's gonna this over gold?
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Right? Like, there is a problem.
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Why Bitcoin?
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This is the why.
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What makes Bitcoin hard
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is not this it's not so much the supply cap,
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because the supply cap can be changed.
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Right?
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It's
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a question of what makes it hard and it's cryptographic
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hardness
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that I think just yeah, we think goes a long way into building the conviction that's going to be needed
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when man, it's going to get hard out there. It's one of these things that maybe a year from now I'll be able to talk about more, but like scanning code bases and finding broken things and then spending tens of thousands of dollars to look at libsecp.
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And like the small, like the amount of shit you had to till through to find the smallest little thing that was like an optimization
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versus like cursory levels of like scanning and finding things that are worse.
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Right?
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It was very cool. Like, it was very
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validating of a thesis
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for sure. Yeah. And not that I wanna do podcast planning while we're here right now, but I would say that I think it makes sense to do at least one episode where you bring
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to the extent you can, where you bring some of your learnings from because you are uniquely in there.
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Right? And so maybe there's some learnings you wanna bring back to the to the podcast. Yeah.
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For sure. That that would be fun. The other there's actually two cryptographic
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things that actually really matter in Bitcoin. There's two
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foundational concepts, and we've been teasing this for a while. The first one is libsecpi and the signatures.
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The second one is shop too.
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So I know we
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need to get over the hump and just start doing hash functions.
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We're getting that and I thought about
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starting that today, but I don't
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feel like we are ready to
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like if it becomes a 10 episode arc, we're just not I don't feel like we're ready. No. I don't think we're ready. And I think this is we're just on sabbatical. Right? So we started the podcast. We did low sec pee for 10 episodes. Now we're just kind of like, it's summer break. We're just kind of vibing. Summer breaks on the wind down. We're, you know, we're gonna get through the rest of it. We'll hit the ground running soon enough.
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Yes.
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But you got we were saying Shah two.
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Yeah. But SHA two is the other I mean, there's also there's also RipeMD one sixty,
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and that's used for,
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what's called hash one sixty where you take data, you take like a public key, you SHA two fifty six hash it, then you write them d one sixty hash it, and then you encode that into a Bitcoin address.
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There's a special opcode that just does that. It's called hash one sixty because it's SHA two hash plus the one sixty.
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We could talk about all that. And then we'll have a good little like maybe sidebar too about like why SHA one is now broken,
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which got broken in the past couple of years.
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Document
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does cover all of this.
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So
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not due to a lack of covering the low hanging stuff. Right? No, no, no. I mean, think it's okay to talk about right now. Swear I find it a little bit
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like what hasn't clicked for me yet,
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and maybe we'll just start until it clicks. But what
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clicked for me is how you actually create a hash function. Like how the heck like, you know, I want to be able to I want the algo to click. Like I want to be able to visualize the
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what's called the avalanche effect.
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Want to be able to visualize.
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Yeah, and I feel like I had, normally
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going through a document like the one I made, the workbook that we'll do,
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gets me pretty close to just being able to visualize the key thing.
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It worked with SecPeek.
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Like we got through, I feel like we got through it and everything that needed to click clicked.
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Right. And with the hash function, it's maybe one of these things, it's maybe unclickable to really be able to see
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how it does what it does, which is maybe what Alan was telling us on our episode.
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Yeah. And he's been trying. He's
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been
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hankering away
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at it and having the same sort of the same result. Right? This may be one of those real black box.
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It's just a lit like a little bit of a black box of the hash function I'm still not satisfied with
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to feel like, you know what I mean?
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I will say SHA two is more vibes than elliptic curve cryptography.
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Yes. And that's so maybe this is because it's it's like Yeah. It's it's basically like we're doing sufficient shuffling. This is good. Whereas, like, you don't have like a closed group set
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that you're doing modular arithmetic across a generator. You know what mean? Like, it's just inherently a different But
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it's yeah, it's different. So the key difference from
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an algebra perspective is that
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the elliptic curve is a group,
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right? And
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we, again, yet to be proven, but it's an isomorphism
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between
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the points in the elliptic curve with the operation of addition
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and the finite field.
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Right? The finite field modulo
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big p.
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I wonder if this is one of these things that you could load into,
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like a lean proof and send a bunch of agents out it and just formally prove it and be done. I think we wanna do that because
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the hash function is not a group. It's a semi group. It's called the semi group. And why it's not a group, why it's a semi group is
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because it doesn't have the inverse
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By design. Absolutely.
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By design, That's
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whole point. Exactly, exactly. So The whole point is you can't inverse it. Maybe
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we just needed a little bit of this abstract algebra
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kind of context so that we can just
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feel good going into whatever the whatever vibes we end up having to accept
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Right. On
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hash functions is going to be just due to the fact that it's not a actual group. It's designed to be what's called a semi group. And
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weirdly enough, you know, something that's less of a group you'd think would be easier to understand, but this is like graduate level that's like graduate level
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algebra.
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And,
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you know, it might be again, we just try to broad stroke this thing out.
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By
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design it has no inverse, and that's literally what a hash function is, it's deterministic.
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It's so interesting that it's deterministic too, right? Because it's kind of associated with
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this low probability thing, but it's totally
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deterministic.
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But
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it's just that the ability to go backwards has been destroyed.
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Whereas with a group, you need to go backwards, otherwise you're fucked.
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It's the whole right. So I like this, like I I think people who are tuned into the Magic Internet podcast are
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they're
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happy that we're just talking podcast business on the podcast because maybe they can help steer us in the right direction for how to tee up this conversation anyway.
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Are there any other
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current events, math things that are grinding your gears you wanna talk about? Anything from Midwest Bitcoin Summit?
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Anything you wanna share for the class?
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Yeah, why not? Let's go into Let's get into As
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somebody who people love talking about math with when they meet me.
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I'm gonna shout out this guy Jim Carucci
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who
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is a fun fucking dude. I met him, befriended him. You know, he's one of those guys. I kinda connected with him because he just
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he's like 20 younger than me, but he reminds me of, like like, these kids I grew up in Jersey with. He's got that new that just that kinda New York energy, that swag.
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I kinda the way I met him was he would get into my he would be like this guy that replies replies to to you you on on Twitter. Twitter. But if you don't know him, you'd think like he's kind of like this aggressive, like,
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dude, what the fuck? Like,
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it sort of sounds like these gotchas,
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like, it just sounds he seems aggressive on Twitter when you don't know him. Then when I met him so then I met him last week and I'm like, dude, I get it. I fucking get this guy. He's just an intense
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he's just an intense,
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dude.
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Anyway, so he
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spent a good amount of time
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talking to me about Lewis Carroll,
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author of Alice in Wonderland.
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Mhmm. And he's been in a I mean, rabbit hole would be an ironic term for this, I guess.
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But,
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but apparently Lewis Carroll
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was a
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hater,
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absolute hater of imaginary numbers and devoted a good portion of his life to either invalidating them,
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telling people why they're gay,
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or trying to get and he was a mathematician.
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Apparently, he was also a mathematician.
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So like we spent a good like I was blown the fuck away by this because it made like my god, the original rabbit hole was Lewis Carroll being like you remember we talked about the Pythagoreans?
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Yes. You know, and it's like, okay, so Pythagoras didn't believe in rational numbers and then one of his own people used his own formula
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to like
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prove it and they had to kill him. They had to kill the guy, lest Pythagoras discover that his worldview collapsed.
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So like Lewis
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Carroll of Alice in Wonderland had this worldview that
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imaginary numbers couldn't exist.
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It goes deeper than that.
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Was basically saying that Alice in Wonderland itself
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was a rebuke
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against quaternions
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somehow. Okay. Do you know what quaternions are?
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Off handout.
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Oh, this is the coolest. To me, this is one of the coolest things ever in the world, and I'm so glad we have
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as many episodes of abstract algebra and cyclic groups behind us so that I can just really show you. It's a cyclic group,
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00:20:48.835 --> 00:20:52.595
right, that's generated by the imaginary number I.
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So it has four members,
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okay, it has I,
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And then you say, okay, you take I squared, so it says I and negative one.
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And then it goes to negative I.
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Yeah. And then it goes back to one.
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And imagine
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a
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graph, a Cartesian graph, right? Sure.
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00:21:12.685 --> 00:21:17.005
Where you start, so I would be like, I in numbers,
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the x value is the real value and the y value is the imaginary value.
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So
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I
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would be
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zero, so the identity point would
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be zero, one, which is one, right? One is your identity point, but I is your generator.
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00:21:35.600 --> 00:21:37.679
So one times I is I, so
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imagine you're now going
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90 degrees. This is, every time you multiply by I it's a 90 degree rotation.
321
00:21:45.125 --> 00:21:47.365
Rotation. Yep. Right, so you go from
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one zero to zero one, that's the I point.
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Eye point and that's zero one.
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And then you go to negative eye, sense of now you're at 180 degrees,
325
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right? Negative
326
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eye. Negative one zero. Yep. And then you go back
327
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to
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negative eye, which is the two seventy degree point, right? Then you go back and then you have this sort of period you know, you have this perfectly periodic function.
329
00:22:14.415 --> 00:22:17.934
That's the quaternion group is those four those four members.
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Got it. And there's
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that's
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a very shallow introduction
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to it. If
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you study like complex analysis, Kayla is studying complex analysis right now and it's like,
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00:22:32.940 --> 00:22:38.674
if you do even more foundational work in like real analysis, it's like what
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00:22:38.835 --> 00:22:41.075
opens up here is insane
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and very exciting. And it's almost
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maddening to think that
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this guy who's known as one of the more creative authors
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we've had whose work still stands through time
341
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and is so relevant
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could have somehow for whatever reason been such a hater and disbeliever or whatever of this. It
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makes no sense.
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People
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this is
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an observation of just human psychology that people
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create an identity and an ego around a set of ideas, and they're unable to separate their own sense of self from the ideas.
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And from that,
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you kind of depart from reality and the laws of nature and it becomes a cult of personality, and that's where you just have stuff like killing people who believe in imaginary numbers.
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00:23:42.115 --> 00:23:43.635
Yeah. Now I should say
351
00:23:44.659 --> 00:23:50.499
it's not like a 100% known to be true that that's the reason he was killed.
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Sure. You don't know for sure. Again, verify, it's hard to go back. We can't ask them.
353
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Yep. We can't ask the Right?
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00:24:00.345 --> 00:24:01.705
It's pretty, know,
355
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Lies up nicely.
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00:24:03.865 --> 00:24:05.065
It's good lore.
357
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Okay? Absolutely. And it tracks, so it's good lore. And
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so Jim was sending me these clanker interactions he was having after the conference. The conference got home two days ago and he started sending me all this back and forth with clanker
359
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around
360
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just the whole Lewis Carroll thing, right?
361
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And, you
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00:24:30.565 --> 00:24:40.980
know, it's a bit questionable whether or not Alice in Wonderland was truly about that, but Lewis Carroll was a legit hater of imaginary numbers. And I, you know, like, I kinda get
363
00:24:41.540 --> 00:24:45.220
like, I can see that in the time. Like, I can see somebody be like, what the fuck?
364
00:24:45.620 --> 00:24:50.100
Yeah. I'm not I'm not down with imaginary numbers. I don't know. In fact, I don't even like the framing.
365
00:24:51.655 --> 00:24:57.575
I like the framing of like, how do you solve x squared plus one equals zero? Like, I like that framing.
366
00:24:59.495 --> 00:25:00.455
You need some Right.
367
00:25:01.575 --> 00:25:03.495
Right. There needs to be
368
00:25:04.009 --> 00:25:07.610
a thing that answers that question. Now maybe was imaginary
369
00:25:07.610 --> 00:25:10.329
a poor choice of words maybe?
370
00:25:11.690 --> 00:25:13.449
Was that a poor choice of words?
371
00:25:14.250 --> 00:25:18.409
Do you think, and here's something, imaginary
372
00:25:19.455 --> 00:25:24.254
seems to be, if in today's day and age, seems to be a poor choice of words, right?
373
00:25:24.815 --> 00:25:25.695
Like oh,
374
00:25:26.255 --> 00:25:28.414
and we have real and imaginary,
375
00:25:28.414 --> 00:25:28.894
is
376
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weird.
377
00:25:31.500 --> 00:25:32.540
And
378
00:25:32.540 --> 00:25:36.620
part I wonder if I actually wonder if the word imaginary
379
00:25:36.860 --> 00:25:38.620
comes from the term image.
380
00:25:41.500 --> 00:25:42.620
Imaginary means I
381
00:25:43.020 --> 00:25:44.300
don't know the etymology offhand.
382
00:25:44.674 --> 00:25:45.394
Well,
383
00:25:45.475 --> 00:25:47.154
in math is like the
384
00:25:47.794 --> 00:25:51.715
in math, so you know what a pre image is. So again, like a hash You
385
00:25:53.635 --> 00:25:56.754
have a function is like a machine, right?
386
00:25:57.075 --> 00:25:59.715
It's a map that takes an input
387
00:26:00.070 --> 00:26:00.710
and
388
00:26:01.590 --> 00:26:04.950
does something to it and creates an output, right?
389
00:26:05.510 --> 00:26:07.590
And that output is called the image.
390
00:26:08.310 --> 00:26:12.950
The input would be called the pre image, right, that causes the image, right?
391
00:26:13.725 --> 00:26:16.605
You might call it a domain, right? The output,
392
00:26:17.005 --> 00:26:24.205
maybe the domain is like the set of numbers that can be a pre image and the range is like the set of numbers that can be
393
00:26:25.485 --> 00:26:27.404
an image, right?
394
00:26:27.725 --> 00:26:32.499
But like the image is like the f of x, so if I have a function f,
395
00:26:32.659 --> 00:26:36.419
and I input x and I get f of x, f of x is my image.
396
00:26:36.980 --> 00:26:37.859
The eye,
397
00:26:38.100 --> 00:26:42.980
that's why having the eye be on the y axis, the y axis is where the image is plotted.
398
00:26:44.025 --> 00:26:47.145
Right? The x axis is where the pre image is plotted.
399
00:26:48.185 --> 00:26:51.705
And I wonder if imaginary, it just refers to the word image
400
00:26:52.345 --> 00:26:54.425
and not meant to be some mystical
401
00:26:54.505 --> 00:26:59.145
some mystical explanation that for all time got people to say, well, I don't have to care about that shit.
402
00:27:00.770 --> 00:27:01.970
It's imaginary.
403
00:27:02.850 --> 00:27:04.610
It's not even real. What the fuck?
404
00:27:05.010 --> 00:27:05.730
Right.
405
00:27:06.690 --> 00:27:07.570
I don't know. Right.
406
00:27:13.385 --> 00:27:20.105
Maybe we'll do an imaginary number segment after we get through hash functions. I'm not sure if there's much imaginary numbers though in the applied math of
407
00:27:20.345 --> 00:27:22.025
what we're using in cryptography.
408
00:27:22.025 --> 00:27:24.184
I'd have to go poke around to see if it's used elsewhere.
409
00:27:24.760 --> 00:27:30.280
I don't think so but that the quaternion group I mentioned is definitely isomorphic
410
00:27:30.280 --> 00:27:31.799
to a particular
411
00:27:32.760 --> 00:27:33.879
finite field.
412
00:27:34.440 --> 00:27:34.919
Probably
413
00:27:36.520 --> 00:27:38.360
a finite field modulo
414
00:27:41.395 --> 00:27:42.274
three
415
00:27:42.515 --> 00:27:44.274
is big. There's
416
00:27:44.274 --> 00:27:47.634
no way it's not true. Someone we're gonna get somebody to rage.
417
00:27:47.875 --> 00:27:51.955
We can rage, calmly. That's what we do. I mean, it's definitely there's the hip.
418
00:27:52.750 --> 00:27:53.710
That field,
419
00:27:53.790 --> 00:27:56.510
that the quaternion field is definitely isomorphic
420
00:27:56.510 --> 00:27:57.309
to something
421
00:27:57.549 --> 00:28:00.509
that we're familiar and that so that doesn't operate
422
00:28:01.150 --> 00:28:03.950
with imaginary numbers. It doesn't operate in a complex number system.
423
00:28:05.345 --> 00:28:07.905
When you and that's the whole point of isomorphic
424
00:28:07.905 --> 00:28:08.625
groups
425
00:28:08.705 --> 00:28:09.585
is that
426
00:28:10.065 --> 00:28:15.665
sometimes you want to work with one, sometimes you want to work with the other, but they're
427
00:28:15.665 --> 00:28:16.865
essentially the same
428
00:28:17.470 --> 00:28:19.629
from the perspective of they preserve
429
00:28:20.110 --> 00:28:21.389
the operations,
430
00:28:21.710 --> 00:28:23.389
right? So they preserve
431
00:28:23.549 --> 00:28:26.029
addition and multiplication in your operations.
432
00:28:26.750 --> 00:28:27.789
Or in
433
00:28:28.029 --> 00:28:29.630
the case of a group, it preserves
434
00:28:29.630 --> 00:28:30.110
the operation.
435
00:28:30.885 --> 00:28:33.044
For the leptic curves, it preserves
436
00:28:33.765 --> 00:28:34.804
point addition
437
00:28:35.045 --> 00:28:37.765
the same way the finite field preserves the modular
438
00:28:40.085 --> 00:28:40.405
addition.
439
00:28:42.900 --> 00:28:43.940
So what
440
00:28:44.900 --> 00:28:46.100
an isomorphism
441
00:28:46.100 --> 00:28:46.980
is, is
442
00:28:47.300 --> 00:28:52.900
essentially two groups, two things are equal, completely equal, even though they look very different.
443
00:28:53.380 --> 00:28:54.820
But they are equal from
444
00:28:55.220 --> 00:28:57.700
the algebraic perspective that they preserve the operation
445
00:28:59.975 --> 00:29:00.614
and
446
00:29:02.934 --> 00:29:04.054
there's this,
447
00:29:07.335 --> 00:29:09.815
they're one to one and onto. So basically every
448
00:29:10.054 --> 00:29:11.654
point corresponds to another point,
449
00:29:12.350 --> 00:29:13.950
right? Every point in the
450
00:29:14.910 --> 00:29:16.990
domain corresponds to a point This in is
451
00:29:17.470 --> 00:29:17.870
classic
452
00:29:18.990 --> 00:29:22.510
like being a function where you can't have two different
453
00:29:22.510 --> 00:29:22.990
inputs.
454
00:29:24.705 --> 00:29:26.065
What is it? You can't have
455
00:29:27.585 --> 00:29:29.264
with one to one and onto.
456
00:29:29.664 --> 00:29:30.225
I'm
457
00:29:30.945 --> 00:29:33.184
reaching back now because it's been a couple years.
458
00:29:33.585 --> 00:29:34.065
When
459
00:29:34.705 --> 00:29:35.825
you have a function
460
00:29:36.145 --> 00:29:42.440
you need to have the one function always and the output needs to always have one outcome.
461
00:29:43.240 --> 00:29:47.799
That would be one to one and that's one to one. So like
462
00:29:48.840 --> 00:29:50.279
y equals x,
463
00:29:51.080 --> 00:29:51.719
right?
464
00:29:52.039 --> 00:29:56.895
Sure. F of x equals x is a good example of a function, right?
465
00:29:57.295 --> 00:30:00.575
Right. Where every input has a unique output.
466
00:30:01.935 --> 00:30:02.575
Right?
467
00:30:02.975 --> 00:30:03.695
No but
468
00:30:05.455 --> 00:30:08.255
then y equals x squared is
469
00:30:08.895 --> 00:30:12.940
a function. F of x equals x squared is a function where that's
470
00:30:14.460 --> 00:30:17.900
not necessary. Well, it's true from every input
471
00:30:17.900 --> 00:30:21.500
has a unique output, but it's not Every output doesn't have a unique input.
472
00:30:21.660 --> 00:30:23.820
Exactly, right. So it's not onto.
473
00:30:25.515 --> 00:30:27.115
Onto is sort of like
474
00:30:29.595 --> 00:30:32.235
every point in the range is represented
475
00:30:33.275 --> 00:30:33.995
somehow
476
00:30:34.875 --> 00:30:46.239
by this function. This function covers every single point in the range. And I'm about to bring this back to hash functions. Hash functions are good if they're onto, but the moment they're no longer onto,
477
00:30:46.799 --> 00:30:49.119
it breaks because you could have two
478
00:30:49.679 --> 00:30:52.640
every x can have a y, but the moment you have
479
00:30:54.425 --> 00:30:57.465
two different Ys and you can't understand
480
00:30:57.465 --> 00:31:04.105
what X it went back from two because there's multiple because you have a collision, it's no longer onto, which means that it's a requirement for a good hash function.
481
00:31:04.840 --> 00:31:05.720
Yes.
482
00:31:05.720 --> 00:31:09.560
And I guess as you're talking, I'm just intuiting that like, it's
483
00:31:09.560 --> 00:31:13.880
probably true that hash functions are not one to one because there
484
00:31:14.200 --> 00:31:18.040
almost have to be collisions for the inverse not to exist. Definitionally,
485
00:31:18.040 --> 00:31:21.415
has to be collisions because if you're collapsing to a 256
486
00:31:21.415 --> 00:31:24.695
bit space for SHA two, there's more than 256
487
00:31:24.695 --> 00:31:27.494
bits of information you could put out into the function.
488
00:31:27.735 --> 00:31:30.134
Because if if a hash function, you could take any
489
00:31:30.775 --> 00:31:39.730
ultimately, what it actually is, it's any number. Right? What you do is you take a data representation, you make it hexadecimal and base 16, all of a sudden you have a number. So you can make any file,
490
00:31:39.970 --> 00:31:48.585
anything a number. You just encode it into a number. And then once you encode it into that number, you compress it through a space that's two to the two fifty six.
491
00:31:49.145 --> 00:31:50.265
Necessarily,
492
00:31:50.825 --> 00:31:59.385
if you have more than two fifty six bits of information you could put into the function, which you can, you can put anything into this and you could just take
493
00:32:00.265 --> 00:32:05.920
from like a you could just say, I'm going to pick every number between zero and two to the two ten
494
00:32:05.920 --> 00:32:06.799
twenty four.
495
00:32:07.440 --> 00:32:08.400
The
496
00:32:08.400 --> 00:32:10.799
problem is that it's just such a large number.
497
00:32:11.679 --> 00:32:16.960
You'd have to wait for the heat death of the universe with the math that we have today to be able to check all of the permutations of that.
498
00:32:18.025 --> 00:32:31.385
And also at that point, you're way more effective to just go reverse engineer the elliptic curve than you are to try and map out all of the hash function outputs, right? Like it's just, if you're trying, if you're gonna run this compute for something, you're gonna run it more so
499
00:32:32.610 --> 00:32:35.250
for the private keys than you are for the hashing. But this
500
00:32:35.970 --> 00:32:39.010
is also from Bitcoin mining. It's
501
00:32:39.010 --> 00:32:44.210
kind of cool when you think about it this way. You take every single Bitcoin block and you convert it into a number.
502
00:32:45.215 --> 00:32:47.615
And the Bitcoin block specifically,
503
00:32:48.655 --> 00:32:53.934
it's cool, like, as a small tangent, like how Bitcoin block headers are constructed. You put a Merkle root.
504
00:32:55.215 --> 00:33:03.930
We probably should do it like we we could do like a a one off episode just talking about Merkle trees. Like you can By do Merkle root the way, have a study guide for Merkle trees. I have a whole study It'll guide for
505
00:33:03.930 --> 00:33:06.570
be a nice introductory way to talk about all the hashing stuff.
506
00:33:07.370 --> 00:33:14.490
But in the block header, you put the Merkle root of all the transactions that are in that block in there. So miners don't have to load the
507
00:33:15.025 --> 00:33:17.745
miners when they are doing their mining operations.
508
00:33:18.065 --> 00:33:20.945
They're not every single time they're turning the hash function,
509
00:33:21.185 --> 00:33:37.240
they're not putting the entire block through. They just put the block header in there. But the block header represents all of the transactions within that block. So it's a nice, like, compression for just doing the work to turn to do the guessing for the hash functions. But you're ultimately just taking you're compressing all of the information into a number.
510
00:33:38.120 --> 00:33:41.400
And you then turn it through the hash function, you get out to the other side.
511
00:33:44.375 --> 00:33:52.855
I always like going to sped school. I always like thinking of Merkle trees and Merkle roots as like finding and being able to find a needle in the haystack
512
00:33:53.495 --> 00:33:55.255
because we can track the needle.
513
00:33:56.320 --> 00:33:56.960
Right.
514
00:33:57.280 --> 00:34:04.320
And then the Merkle proof, you could actually reveal the chain of the Merkle tree to show you have this one bit of data anchored into
515
00:34:04.640 --> 00:34:05.520
hash,
516
00:34:05.520 --> 00:34:06.880
but then you can provide
517
00:34:06.960 --> 00:34:07.760
basically
518
00:34:08.240 --> 00:34:13.735
the whole chronology of getting back to the root original data. So back to the hash function.
519
00:34:15.575 --> 00:34:17.655
Cool sort of just a cool distinction
520
00:34:17.655 --> 00:34:19.335
that's different between
521
00:34:20.775 --> 00:34:22.855
the elliptic curve and the hash function
522
00:34:23.175 --> 00:34:32.630
is this goes to the fact that there's no inverse, right? And when I say there's no inverse, it's like it's either the inverse is either nonexistent or non unique. And I'm not
523
00:34:33.030 --> 00:34:37.190
well versed enough. I know it's not unique because that means there are collisions
524
00:34:37.190 --> 00:34:38.070
necessarily,
525
00:34:38.070 --> 00:34:39.670
and so the inverse is not unique.
526
00:34:41.190 --> 00:34:41.510
Anyway,
527
00:34:42.045 --> 00:34:48.765
how it works, we know about the so with the elliptic curve, we know about the finite field is modulo
528
00:34:49.724 --> 00:34:50.605
prime,
529
00:34:51.005 --> 00:34:51.965
correct? Yep.
530
00:34:52.365 --> 00:34:55.885
And it has to be modulo prime and that's what ensures
531
00:34:55.885 --> 00:34:57.165
this isomorphism.
532
00:34:57.750 --> 00:35:02.230
This one to one, this onto, this ability to always go back and forth
533
00:35:02.470 --> 00:35:03.350
between
534
00:35:05.030 --> 00:35:11.590
spaces, the elliptic curve and the finite field. It's the fact that it's modulo prime is what allows it to do that, right?
535
00:35:12.665 --> 00:35:14.025
In a hash function,
536
00:35:14.505 --> 00:35:17.705
the modulus is two to the two fifty six. The modulus is,
537
00:35:18.984 --> 00:35:23.545
at least for SHA-two 56, it's two to the two fifty six, which we know is not prime.
538
00:35:24.185 --> 00:35:26.744
And so that's part of the design.
539
00:35:27.830 --> 00:35:28.710
It's called
540
00:35:29.430 --> 00:35:29.910
the
541
00:35:30.870 --> 00:35:35.510
Galois field. Galois fields are where the modulus is two to the something,
542
00:35:35.990 --> 00:35:36.630
to n.
543
00:35:37.350 --> 00:35:47.355
So it's interesting that that's part of the design of how there's no inverse and part of how it's a semi group is the fact that the modulus is not prime, it's two to the two fifty six.
544
00:35:47.835 --> 00:35:48.475
Whereas
545
00:35:48.635 --> 00:35:52.474
in the finite field for the elliptic curve, it's two to the two fifty six minuteus
546
00:35:52.474 --> 00:35:55.675
two to the thirty second minus nine seventy seven. It's
547
00:35:55.994 --> 00:35:56.555
prime
548
00:35:56.795 --> 00:36:00.080
it's the prime number Satoshi wanted to make that work.
549
00:36:01.840 --> 00:36:03.440
To make elliptic curve math
550
00:36:04.160 --> 00:36:05.280
work for us.
551
00:36:06.800 --> 00:36:09.040
But in the hash function it's not the case.
552
00:36:10.160 --> 00:36:15.815
Finite fields which we haven't really done a lot on when there's not a prime modulus, but it's
553
00:36:15.895 --> 00:36:16.935
a little haywire.
554
00:36:17.415 --> 00:36:19.655
It doesn't behave quite as nice.
555
00:36:21.095 --> 00:36:22.135
Exactly.
556
00:36:23.015 --> 00:36:23.335
Exactly.
557
00:36:24.980 --> 00:36:26.900
Yeah. It's kinda And I
558
00:36:27.300 --> 00:36:33.140
would in terms of one to one and onto, since we're on the topic, I was gonna offer I
559
00:36:33.460 --> 00:36:38.055
was gonna offer the linear algebra. This is like, I'm gonna spoil
560
00:36:38.055 --> 00:36:41.255
something for you guys because when this clicked for me, was
561
00:36:42.135 --> 00:36:45.015
like seeing the sun and the stars align
562
00:36:45.735 --> 00:36:51.380
and one of the most amazing experiences actually of my life. But it's unlikely that you guys are gonna
563
00:36:52.099 --> 00:36:55.460
you're gonna go for that experience. So I'm gonna actually do this. So
564
00:36:55.859 --> 00:37:00.260
linear algebra fans. Okay? You guys know you know what linear independence
565
00:37:00.500 --> 00:37:01.940
means? I just take it.
566
00:37:04.345 --> 00:37:05.625
Linear independence
567
00:37:05.625 --> 00:37:06.665
being the
568
00:37:09.944 --> 00:37:16.025
you're gonna have to you're gonna say So linear independence means remember it, but that they're like in a system
569
00:37:16.490 --> 00:37:17.850
that there's no
570
00:37:22.010 --> 00:37:22.890
there's no
571
00:37:23.130 --> 00:37:27.450
there's no zero like the output is not zero unless all the inputs are zero.
572
00:37:29.085 --> 00:37:34.925
Yeah. I mean, I'm cheating. Just like if if no vector in the set can be written as a combination of the others, which is definitionally
573
00:37:34.925 --> 00:37:36.205
like orthogonal.
574
00:37:36.525 --> 00:37:38.205
Yes. And that means that it
575
00:37:39.405 --> 00:37:40.445
has a basis.
576
00:37:40.765 --> 00:37:43.645
Right? So that's one to one We're gonna go to
577
00:37:44.550 --> 00:37:47.750
this is on the docket too. We just need to do like a whole little
578
00:37:49.270 --> 00:37:52.950
mini series on linear algebra. I get Yeah, we definitely should. In there.
579
00:37:54.390 --> 00:38:00.615
I think that'd be a fun one. That's also LLM, Jason. So like that I should say prior in iteration
580
00:38:00.615 --> 00:38:01.895
of this podcast,
581
00:38:02.295 --> 00:38:05.175
Motivate the Math with Gary, shout out Gary Krausz.
582
00:38:05.975 --> 00:38:13.575
The I saw Gary for like ten minutes at the BPI Summit. Nice, nice. Gotta get us all on and do a full do
583
00:38:14.000 --> 00:38:15.280
Anyway, there
584
00:38:15.520 --> 00:38:17.520
was an episode I think called where
585
00:38:17.520 --> 00:38:23.520
we talked about basis and standard basis because I was basically saying that Bitcoin is a standard basis for money.
586
00:38:25.119 --> 00:38:26.960
And we never had a standard basis.
587
00:38:28.285 --> 00:38:30.925
The standard basis, think about in
588
00:38:30.925 --> 00:38:33.325
the Cartesian x y plane, right?
589
00:38:33.725 --> 00:38:35.565
You have x axis,
590
00:38:35.885 --> 00:38:38.685
I. E. Zero, one or zero x,
591
00:38:39.165 --> 00:38:39.805
but in
592
00:38:40.285 --> 00:38:44.200
an elementary basis, zero, one and the point zero, one,
593
00:38:45.160 --> 00:38:53.560
and then in the y axis, the point one, zero, those form a basis, a standard basis. Like an elementary basis, right? And
594
00:38:53.880 --> 00:38:59.685
everything inside that plane is a linear combination of those two things, right? But the only way to
595
00:39:01.444 --> 00:39:04.244
get the answer of the linear combination to be zero
596
00:39:04.325 --> 00:39:06.325
is if both x and y are zero.
597
00:39:06.644 --> 00:39:07.605
And
598
00:39:07.605 --> 00:39:09.684
that is the essence of one to one,
599
00:39:10.580 --> 00:39:12.820
right? So like if f of x
600
00:39:13.700 --> 00:39:14.500
equals
601
00:39:14.500 --> 00:39:19.620
f of y equals if f of x equals f of y, then x has to equal y.
602
00:39:20.020 --> 00:39:21.380
That's
603
00:39:21.060 --> 00:39:21.860
means, what one to
604
00:39:22.745 --> 00:39:23.705
right? So in
605
00:39:24.345 --> 00:39:28.425
linear algebra, this idea of linear independence gives you one to one.
606
00:39:29.065 --> 00:39:30.505
And then there's
607
00:39:30.505 --> 00:39:33.145
another concept in linear algebra called spanning the set.
608
00:39:35.150 --> 00:39:38.590
These are the most these are, like, the most fundamental
609
00:39:38.590 --> 00:39:42.110
concepts of now if you took linear algebra in college,
610
00:39:43.950 --> 00:39:45.470
it was probably misnamed.
611
00:39:45.790 --> 00:39:47.950
You probably took a course called matrices,
612
00:39:48.875 --> 00:39:53.915
where you learnt the arithmetic of linear algebra, right? Where you learnt how to reduce a matrix,
613
00:39:54.395 --> 00:39:57.035
you learnt how to calculate an eigenvalue,
614
00:39:57.435 --> 00:40:02.234
you see what I'm saying? Whereas what I'm talking about, linear algebra is the abstract,
615
00:40:03.030 --> 00:40:05.830
it's more of the abstract essence of these
616
00:40:05.830 --> 00:40:09.750
ideas of linear independence and spanning the set are abstract concepts.
617
00:40:10.790 --> 00:40:13.830
So spanning the set is what we said before, meaning like
618
00:40:16.645 --> 00:40:17.605
cover every
619
00:40:18.725 --> 00:40:19.445
input,
620
00:40:19.685 --> 00:40:21.925
your function covers every output
621
00:40:22.325 --> 00:40:25.045
that's possibly in the set. So in other words,
622
00:40:25.445 --> 00:40:35.250
if you have y equals x squared and your output, your range space is all the real numbers, then it doesn't span the set because x squared is only going give you positive numbers.
623
00:40:36.130 --> 00:40:40.130
Zero. You're never going get negative numbers out of that set, so it doesn't span the set.
624
00:40:41.055 --> 00:40:41.775
If your
625
00:40:42.175 --> 00:40:44.175
output is all the positive numbers
626
00:40:45.214 --> 00:40:47.775
and zero, then it does span the set.
627
00:40:48.175 --> 00:40:49.855
That's what onto means.
628
00:40:50.175 --> 00:40:51.855
So onto just means that
629
00:40:52.335 --> 00:40:54.494
my inputs cover the entire set.
630
00:40:56.819 --> 00:40:58.980
One to one means that
631
00:40:59.220 --> 00:41:00.020
my
632
00:41:00.339 --> 00:41:02.900
inputs and outputs are linearly independent.
633
00:41:03.140 --> 00:41:04.900
If those two things are
634
00:41:05.059 --> 00:41:06.420
both there at the same time,
635
00:41:11.515 --> 00:41:13.275
and my function preserves
636
00:41:13.595 --> 00:41:15.035
the operation,
637
00:41:15.755 --> 00:41:16.954
you have a
638
00:41:17.515 --> 00:41:19.755
magical thing called an isomorphism
639
00:41:20.420 --> 00:41:28.980
where two things don't look like they're the same at all, right? Like XOR and the finite field modulo two.
640
00:41:29.940 --> 00:41:31.140
These are isomorphisms.
641
00:41:31.140 --> 00:41:33.140
These are the same thing. They're just different labels.
642
00:41:34.805 --> 00:41:47.525
Two things don't look alike. And why does that matter? Is because one thing might actually possess mathematical properties you can understand and work with and one might not. And so you can actually transport yourself
643
00:41:47.924 --> 00:41:49.285
into dealing with something
644
00:41:49.900 --> 00:41:51.260
in equivalence
645
00:41:51.420 --> 00:41:52.940
that
646
00:41:53.420 --> 00:41:56.860
actually gives you power over understanding it. I
647
00:41:57.020 --> 00:42:04.780
look, I think looking at an elliptic curve under point addition is much more powerful to understanding cryptography than looking at that finite field.
648
00:42:05.545 --> 00:42:06.185
Right.
649
00:42:06.745 --> 00:42:09.785
Okay, right? It's like one of them actually opens your eyes.
650
00:42:10.105 --> 00:42:10.665
Now,
651
00:42:11.625 --> 00:42:16.745
goes to why we're here. Like why I'm here is because I'm kind of worried that somewhere down the line,
652
00:42:17.305 --> 00:42:18.105
someone
653
00:42:18.345 --> 00:42:19.865
representing LibSecP
654
00:42:19.865 --> 00:42:20.905
or somebody
655
00:42:21.260 --> 00:42:23.980
in the industry or somebody out of academia
656
00:42:23.980 --> 00:42:27.420
is going to convince us that like the quantum problem,
657
00:42:30.300 --> 00:42:32.220
can work in an equivalent space
658
00:42:32.220 --> 00:42:35.099
and the question is going to be who are we to argue with them?
659
00:42:36.565 --> 00:42:38.005
Who are we going to be to
660
00:42:38.405 --> 00:42:43.045
be able to say yes or no, that that's true. We have to understand the basic
661
00:42:44.405 --> 00:42:50.565
things. So what I'm obsessed with is like whether or not I would agree with somebody that two things are equivalent.
662
00:42:51.000 --> 00:42:53.880
That's why I studied the fuck algebra.
663
00:42:54.840 --> 00:42:57.320
Because I'm obsessed with being able to validate that.
664
00:42:57.960 --> 00:42:58.600
Absolutely.
665
00:42:59.640 --> 00:43:01.880
No, I think It's that simple. When
666
00:43:02.600 --> 00:43:04.440
that time comes, we will be
667
00:43:05.160 --> 00:43:05.800
in a
668
00:43:06.305 --> 00:43:10.065
full emergency podcast. It'll probably be like twice a week,
669
00:43:10.385 --> 00:43:23.410
just like RHR during the pandemic. They did a twice a week podcast. We'll be doing twice a week podcasts looking at like every single thing and just taking everyone through the journey with us and trying to get to our with our own rational reasoning to the other side to help disseminate and distribute
670
00:43:23.730 --> 00:43:26.130
the discourse around this stuff.
671
00:43:26.370 --> 00:43:28.930
So we're gonna have to do a little linear algebra
672
00:43:28.930 --> 00:43:30.370
along the way. Yes.
673
00:43:30.850 --> 00:43:31.730
That's fine.
674
00:43:32.370 --> 00:43:36.050
Right? And then if you guys like AI and you want to code your own local
675
00:43:36.305 --> 00:43:39.505
LLMs, this will be helpful to you. But it's
676
00:43:39.745 --> 00:43:44.145
almost it's very hard to have done what we did skipping through linear algebra.
677
00:43:44.385 --> 00:43:47.745
It's very hard to do what we did. So like we will have to cover that
678
00:43:48.225 --> 00:43:48.865
Yes. Because just
679
00:43:50.799 --> 00:43:59.040
the ethos from my perspective is this obsession of being able to say yay or nay, that two things are equivalent. We're going to be sold,
680
00:43:59.599 --> 00:44:03.119
you know, it's not if it was whatever Project 11.
681
00:44:03.555 --> 00:44:07.795
If it's not them, it's gonna be somebody it's gonna be BlackRock or and, you know,
682
00:44:09.555 --> 00:44:11.875
you know, we're gonna be asking questions like,
683
00:44:12.035 --> 00:44:15.795
did Blockstream guys get bought by these guys? We're gonna be asking those questions,
684
00:44:15.875 --> 00:44:17.635
and and we're gonna have to have some way.
685
00:44:18.420 --> 00:44:21.940
And like I said, the distance between us and that is quite far.
686
00:44:23.940 --> 00:44:24.820
But Yeah.
687
00:44:25.299 --> 00:44:28.579
We're gonna have do our That's
688
00:44:28.579 --> 00:44:34.875
why we decided to do this today even having no great plan. It's like we just gotta keep grinding our grinding to this distance. So
689
00:44:37.115 --> 00:44:37.755
so
690
00:44:37.915 --> 00:44:39.115
linear algebra,
691
00:44:39.195 --> 00:44:40.395
some Merkle tree,
692
00:44:42.235 --> 00:44:42.875
and then
693
00:44:45.435 --> 00:44:50.369
it's a question of, yeah, the hash functions are good because it just I
694
00:44:50.369 --> 00:44:53.330
think for for the most part because it illustrates
695
00:44:54.369 --> 00:44:56.290
the different it illustrates,
696
00:44:56.530 --> 00:44:58.369
like, this isomorphism concept
697
00:44:58.530 --> 00:45:02.850
with what we already with what we are with, like, our base case, which is the elliptic curve.
698
00:45:03.730 --> 00:45:04.849
It contrasts it.
699
00:45:07.785 --> 00:45:10.905
So it'll help build the muscle. It'll help build our muscle.
700
00:45:11.145 --> 00:45:13.385
Right? The categorization of it. Absolutely.
701
00:45:14.425 --> 00:45:18.425
Yeah. It's good to have like a non example. Like this is not an isomorphism.
702
00:45:18.880 --> 00:45:22.080
Right? It's also super important to what we do.
703
00:45:22.960 --> 00:45:24.000
Absolutely. And
704
00:45:24.240 --> 00:45:30.320
people are gonna say, well we need to change this hash function. We're gonna need to, right? And I don't know, that's some people who need to know what that means.
705
00:45:31.535 --> 00:45:35.295
We're gonna change the algo, we're gonna change the hash function, we're gonna change,
706
00:45:35.535 --> 00:45:38.175
like that's the language it's gonna come in.
707
00:45:39.135 --> 00:45:40.895
Hopefully we'll be here running
708
00:45:41.375 --> 00:45:42.575
that, you know,
709
00:45:43.135 --> 00:45:45.615
okay no, there's two primitives,
710
00:45:45.615 --> 00:45:46.895
right? There's the signing
711
00:45:48.160 --> 00:45:49.600
and then there's the hashing.
712
00:45:51.200 --> 00:45:52.160
And they're different.
713
00:45:52.800 --> 00:45:57.760
They're very different. And that's why they're the two critical things. They they they meet
714
00:45:58.000 --> 00:45:59.920
they have different objectives
715
00:45:59.920 --> 00:46:00.880
within the systems.
716
00:46:01.925 --> 00:46:06.485
And maybe we'll be able to live with some of the vibeiness of hash functions because
717
00:46:06.725 --> 00:46:07.445
like
718
00:46:08.005 --> 00:46:12.245
in abstract algebra, we are okay with a lot of information being lost
719
00:46:12.565 --> 00:46:15.845
as long as what is retained is the fact that
720
00:46:16.670 --> 00:46:17.710
the operation
721
00:46:17.710 --> 00:46:18.670
is preserved,
722
00:46:18.830 --> 00:46:20.190
right? And the
723
00:46:21.550 --> 00:46:24.350
one to oneness and the on tooneness, and this
724
00:46:25.070 --> 00:46:28.990
is what we're pointing out. What's important to determine equivalence is
725
00:46:29.310 --> 00:46:30.350
the fact that
726
00:46:31.464 --> 00:46:32.585
it's a
727
00:46:33.065 --> 00:46:33.865
homomorphism
728
00:46:33.865 --> 00:46:37.785
meaning we preserve the operations and we preserve the group properties.
729
00:46:38.025 --> 00:46:39.545
So like that, it's okay,
730
00:46:40.105 --> 00:46:44.905
that's why it's abstract. It's also we use symbols and not numbers to fuck with the GOYS, okay.
731
00:46:47.200 --> 00:46:48.320
You know, it's
732
00:46:49.600 --> 00:46:53.840
abstract partially because we're not interested in every bit of information.
733
00:46:54.000 --> 00:47:00.320
We're not interested in retain you know, what abstract algebra does is says we don't really care what number system you're on. We don't care
734
00:47:01.075 --> 00:47:04.355
about all that one plus one equals two everywhere.
735
00:47:04.355 --> 00:47:07.075
We care that in general, the system preserves
736
00:47:07.155 --> 00:47:10.115
your algebraic operations. These properties. Yes,
737
00:47:10.115 --> 00:47:14.515
preserves these properties. And this is like, I think computer scientists can understand this. Like how do you know,
738
00:47:16.190 --> 00:47:17.550
you inherit properties?
739
00:47:17.550 --> 00:47:19.150
How do you know they're preserved? We
740
00:47:20.910 --> 00:47:22.670
don't care about the entirety.
741
00:47:23.870 --> 00:47:26.590
We care about the mapping, we care about the
742
00:47:28.110 --> 00:47:29.790
one to oneness, the ontuness,
743
00:47:30.455 --> 00:47:31.095
and
744
00:47:31.495 --> 00:47:33.895
whether or not we can really say there's
745
00:47:33.895 --> 00:47:34.855
equivalence,
746
00:47:36.375 --> 00:47:38.215
and if not, why? So
747
00:47:39.335 --> 00:47:45.255
that's all to say is like maybe we shouldn't be too afraid of going through hash functions even if it's
748
00:47:45.255 --> 00:47:48.430
a little black boxed. Because if we can at a high level understand
749
00:47:49.950 --> 00:47:52.750
how it fits into the abstract algebra
750
00:47:53.310 --> 00:47:55.870
equation that we're sort of putting together,
751
00:47:57.470 --> 00:47:58.110
it's good enough.
752
00:48:00.545 --> 00:48:01.265
It's all
753
00:48:04.865 --> 00:48:14.385
these higher level principles are just the things that we need to have comfort and understanding to have integrity in the system. And once you can if you can get those critical pillars understood,
754
00:48:16.170 --> 00:48:22.330
even if you don't want to go into the lower level details, you at least can understand the reasoning behind that. That's a pretty good safety net to lean on.
755
00:48:22.970 --> 00:48:24.570
And I would just make one other
756
00:48:25.050 --> 00:48:30.410
argument is that, so Rob, you're the guy going you happen to be the guy going through libsecpig
757
00:48:30.410 --> 00:48:30.970
with
758
00:48:31.985 --> 00:48:34.225
spending a lot of money One of on many,
759
00:48:34.465 --> 00:48:38.865
one of a couple. Yeah. Understand one of many, but it would be but like arguably,
760
00:48:39.905 --> 00:48:40.385
you
761
00:48:41.025 --> 00:48:43.185
don't
762
00:48:43.185 --> 00:48:50.630
have the full picture of what they ask for still. Oh, trillions of percent. Because we haven't been through, we just haven't really been through all of it.
763
00:48:51.350 --> 00:48:52.070
Be able
764
00:48:52.630 --> 00:48:55.350
to really, really ask these pointed
765
00:48:55.670 --> 00:48:56.470
questions.
766
00:48:57.030 --> 00:48:57.910
Absolutely.
767
00:48:57.990 --> 00:48:59.270
That comes from this
768
00:48:59.830 --> 00:49:00.870
discussion, right?
769
00:49:01.404 --> 00:49:02.125
Yeah.
770
00:49:03.644 --> 00:49:04.204
Agree.
771
00:49:04.845 --> 00:49:06.525
We have some great mathematicians Yeah, doing
772
00:49:07.805 --> 00:49:08.525
for sure.
773
00:49:09.164 --> 00:49:24.369
And there'll be a point probably in the next couple of months where it'll be cool to talk about the little things I did find. Right? Like, and mind you, like, I wouldn't be doing a Bitcoin Magic Internet Math podcast right now if it was all broken. You know what I mean? Like, it's all like, we're good.
774
00:49:25.650 --> 00:49:29.650
Oh, right. You forgot to mention that. You forgot to mention that it was all good.
775
00:49:30.585 --> 00:49:32.185
Yeah, it was good. That
776
00:49:34.105 --> 00:49:39.545
was a very specific strategic thing I did when people are like, it's all Bitcoin and Fugazi.
777
00:49:39.785 --> 00:49:42.265
I started just lipstick P shitposting.
778
00:49:43.050 --> 00:49:47.690
You know, I just think about, like, how how fucking great it was. Like, literally Atlas holding up the world.
779
00:49:49.530 --> 00:49:50.090
And,
780
00:49:50.330 --> 00:49:53.690
it's good. Thank god. It's all good. I know. Oh my god.
781
00:49:55.210 --> 00:49:55.850
Yeah.
782
00:49:56.250 --> 00:49:56.810
Alright.
783
00:49:58.355 --> 00:50:12.355
Do you wanna call up there? We got you got you got anything else? I I got nothing else, Burden. Like, I think it's good. We got a you know, we got we got we got our reps in today. We we we didn't we didn't skip leg day. We came in and we did we didn't skip set day. We came in and we
784
00:50:12.900 --> 00:50:16.980
we did our math. I think this is a good spot. We'll we'll see you again in two weeks.
785
00:50:17.779 --> 00:50:22.740
Yes. Alright. Excellent. Magic Internet Math in the books. See you guys. Peace.