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All right. All right. We are here. Back. Back with the duo. What's up, Rob?

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Hello, hello.

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Welcome back. Magic Internet Math. We are going to, I think this is in episode six.

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Yes.

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We're counting my solo episode, I guess.

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We can count the solo. I think the solo episode is part of the official production.

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Yeah, dude, this isn't like, this isn't pickleball where like if one person can't come, like no one can play.

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You know, the show must go on. We got to keep this going.

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Well, you know, it's really great to be back here with you because now you, you know, right now where we are in the moment, the moment of the show is we have been slogging through the study guide that.

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And, you know, I love, dude, I love going through a math book for me is like every single one I've ever done has a story to it.

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And it's like it's almost like smoking a cigar where like you have the first third, the second third and the last third.

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And each one of them is almost like a different cigar.

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Yes. Yes, very much so.

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If you get through the first third of the math book, the second third is the journey of itself.

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and then if you get really get through that at some point in time in that second third you start

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to say shit i'm actually going to get to the end of this book i'm pretty sure regardless of how hard

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or how much i don't know what i'm even doing i'm just going to get to the end just to just to do it

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right or there's no other way out of this maze you know you hit the point of no return and you

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have to just keep going and that's we're almost at the end of this we're at the end of this

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pretty much i think of the like the arithmetic journey towards starting to kind of now talk

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about the elliptic curve like we've been setting things up i feel like we really did a good job

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with modular arithmetic and inverses it that opened up a lot for me in terms of even just

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explaining my conviction and just so okay this is how we know we know we have this glob thing

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out there called an elliptic curve and i know my private key sits on there and i know i share

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another point out there and i know i need to you know play around with those things publicly to be

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able to sign transactions so but we're like super super zooming in um like that's that's all going

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on on the moon but we're doing things on this earth to try to just to explain why why are we

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looking up there right and so you know modular arithmetic inverse in groups we you know we did

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a good job we can't over repeat groups because that's where you get identities and inverses and

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closure and those things really all base unit that all this stuff built on top of that's right

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and the answer to every question of how do i know like when i manipulate my point on the on the curve

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How do I know I'm still good?

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Right.

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It all comes down to understanding the structure.

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Absolutely.

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You can't calculate it.

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You can't do it on a pen and paper.

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You can't do it with every calculator, every computer on earth.

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Right.

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You need it.

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That's why you need to keep it in an abstract form because you're just talking about the general principle of numbers along this elliptic curve.

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And you can't, to your point, brute force every permutation and give a conclusive proof of all answers.

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Right.

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It's like it's impossible.

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but um and i think it's also important to know the stuff because i do think there are people

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who will manipulate our our lack of willingness to go here it's an easy attack um this is really

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why we do this right now right um so we got through those primitives and i think that we're

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at the we're at this sort of key point here i think it's the big boss of arithmetic it's the

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reason we do all the arithmetic, which is now the discrete log problem, if you think

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we're ready to do that.

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Discrete log?

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We'll go right into that?

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Yeah, I think we're ready to hit discrete.

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I don't think we need to do more on groups.

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I think anything that happens with groups will just continue to come up in repeatable

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fashion.

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I think we're ready to talk about, really, this is the coup de grace right here.

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This is the reason it all works.

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This is the big idea.

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The big idea and this is the thing that people also – it's the topic du jour of the quantums.

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If like, oh, well, if it's discrete logarithm problem, then we're all cooked and all that stuff.

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So we should know what this is.

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This is the key to everything.

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It's more like, OK, if this ever fails, we have to figure out a new one.

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It's one of those – this is the thing that binds – this is the force.

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finds everything together.

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I would say the discrete log problem

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plus proof of work

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are the two

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things that hold Bitcoin together.

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Yeah.

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And the proof of work one is pretty straightforward,

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but I'm going to start there

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because it's an easy principle to understand

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what we're getting at here.

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When

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a Bitcoin miner,

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when they mine for a block

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and they find a block,

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they present the full block saying

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I have solved the puzzle

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and it is very trivial

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for you

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as a node who did not mine the block

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to verify it's correct

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but the amount

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of work to get that answer

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has so much asymmetry on the other side

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where you have to actually just blindly guess

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and

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that's kind of the same idea of the discrete log problem

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where with this public-private-keep

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relationship. It's very trivial if you have the private key to find the public key, but from a

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public key, you cannot trivially reverse engineer and find the private key. That's like the idea.

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Yes. It's this, you know, what you're describing is this two-way pathway where it's very easy to

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walk one, you know, it's very easy to walk one way and it's like impossible to go backwards and

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traverse back to the point you were at it's not possible with it's not that it's not possible it's

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that you know this our civilization hasn't figured out how to create enough power to do it that's

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real and that's so in this regard right what one thing i don't think we've discussed yet here

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is um it's really important in the this whole side of cryptography really works because in a

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sense it's digital physics in a sense it's it's asking this question how much force is required

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to open a door and is it possible with all of the resources that to create force on this earth

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is it actually possible to open a door in order you know how much force is required to do that

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And so that it's the other side of this.

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You know, if you really open to me, it's the very tedious part, but it's also the most important part of these cryptography textbooks.

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Have you like big O notation where we say, OK, this can be solved in the level of N or this can be solved at the level of the square root of N.

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What does that mean? That just means that like, okay, there's like a linear, you know, there's a linear cost or there's a less, like there's a less than linear cost or like there's a more than linear cost.

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I think the simplest way to put it is the computational cost is important to measure.

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And the discrete log problem, in order to solve it, there are ways to solve it that are actually – so the brute force way to solve it takes a certain amount of energy.

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Like 2 to the 128.

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Well, the brute force would be 2 to the 256.

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But because – then there's a couple of – there's been a couple of algorithms like Shanks and the one that – Pollard.

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Pollard's row.

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And by the way, if anybody is looking for a – if anybody is really engaged here and wants a reference, one little book to reference this.

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Neil Koblitz's book called – I think it's called There's Number Theory and Cryptography.

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And that's the Koblitz that's the SEC P256K1.

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It is.

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The K in SECP 256K1 is cobblets.

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Well, yes, but I need to – I've been corrected on this.

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So I'm not ready to publish an errata because I don't know that it's wrong.

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I'm just saying that people have – some people have pointed out that that's not why it's K or that's –

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No, I'm kidding.

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It's a very fun – it's a very actually distinction, but I'm not putting it down because I appreciate the feedback.

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Got it.

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You know, but it is that Koblitz.

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Yes, it's that Neil Koblitz.

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And, you know, as an aside, as an aside, I've talked about this on Motivate the Math because when I started Motivate the Math, I was using his book.

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That was like the book that I decided is his number theory book.

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And it's so good.

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I really recommend it.

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It's small.

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It goes through everything we've been talking about at a pretty good level.

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um but i i had i was sort of so disappointed to hear i saw him on a podcast and you know he does

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work for the ethereum foundation now and he was like this is a guy that invented all this stuff

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and he's going he's on this podcast talking about how privileged he is to be solving like the energy

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problem of the world by supporting a proof of stake um token you know what i mean and it was

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How the mighty fall.

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You see this with mathematicians.

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My like my most kind of the person, the mathematician that's most important to me, my my professor from who opened up everything for me.

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His name is John Alan Paulos, author of the book Enumeracy.

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You know, I follow him on Twitter and I kind of, you know, I know him.

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He used to live.

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He used to literally live in my neighborhood when I grew up.

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It's like I see him on Twitter and he's basically very Kablitzian in his ability to disappoint me and to make me just wonder what – it's a side quest I have, which is – obviously I am of the view of studying math hard is a pathway to personal power.

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But where is the cautionary tale – where does that cautionary tale detour happen also?

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So obviously it's necessary but not sufficient I would say.

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But this is something that I'm thinking – we're talking to people who sort of are already strong people but need this one discipline and would benefit from this one.

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But understand this isn't the answer because you can lose your mind all the time at some point, especially if you're being paid.

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I think you can lose your mind.

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You can lose your way.

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And we see it all the time, the Bitcoin space.

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We're seeing it especially right now in Maxi Madness, not to get too into it.

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Just, you know, seeing the crash outs.

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Anyway, that is all to say Koblitz is alive on this planet and his work, his textbook is excellent.

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Highly recommend it.

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There is a course version on my site based on that book and that's a nice way to access the book too.

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But the book is perfectly accessible and I actually recommend it.

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I like using books if I can.

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Yeah.

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But it really – I feel like it taps me into the thinking.

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It taps me into where they were when they were writing it and I like books.

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So anyway, back to the discrete logarithm problem.

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So we brought up Koblitz because actually he does a really good job of pointing out these costs, the costs of solving these problems.

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Yeah.

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And so it's O to the square root of N is really what the – it's called – there's a giant step.

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There's a baby step, giant step algorithm and then there's Pollard-Rowe.

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And they basically – these are – they elegantly reduce the solving of the discrete log to the square root of N.

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And that's where now it's 2 to the 128 bits.

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But that's still very, very, very, very, very large, right?

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So I think we covered – so we covered the cost.

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The cost is really important even though like – right?

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It's like that is really – because it answers the question, is this safe?

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Can I trust my savings behind this, right?

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right okay uh would you agree we covered that okay i think we covered that as a good

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at a high level um just to maybe uh as a brain exercise for everyone right let's just take a

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small group of like uh mod sets right and let's say three is the generator

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just to keep it in a small space world here um three to the one is three mod seven is just three

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3 squared is 9

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mod 7 is 2

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right 9

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remainder is 2

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mod 2 then you do 3 cubed

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which is 27 mod 7 is 6

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3 to the

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4th is 81 mod

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7 is 4 you can just take my word for this

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or you can do the arithmetic yourself if you want to verify

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3 2 6 4 it's also in the study

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3 2 6 4 and then

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3 to the 5th equals 5

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which is 243

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them on seven and then three to the six goes back to one it wraps back to the start right so we have

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a nice little closed group now if i told you the number was the answer to the question was five

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how many times did i go through the generator you could look at it pretty easily and say you know

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what it's actually five the fifth element of the group is what going through the generators which

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get 2.5. That's a lazy trick.

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Now, this is

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a scaled down example.

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We don't use that as our

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base, but what if the

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list of the group entries is 2 to the 256?

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It was the entire elliptic curve.

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You now have this

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asymmetry

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where

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it's way too large to

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trivially compute all of the answers

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and

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you can have a number

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you start with, which is your private key and put it through the generator.

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And that output gives you the public key.

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But if I just give you that public key, you're not going to be able to do the reverse math

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enough times to be able to find out where that actually belongs.

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So that's like the discrete log problem at a very high level in the application of how

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we use it for public private keys.

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Yes.

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And if you really doubt what Rob's, by the way, everyone listening to this podcast, everyone,

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I know there's a wide range of people listening, but everybody should really go and sit with a pen and paper and do that math and do it also with mod 11, 13 and 17.

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It's not like that hard to do, but you'll start to see how hairy this gets very fast at low numbers.

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And I have a game on the website, magicinternetmath.com.

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I have a game called Mod.

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It's called Mod Racing.

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and it's like you're just trying it's like actually just fast doing module arithmetic in

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your head but this you start to see how hairy this gets pretty quickly and what so what i'm

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going to grab onto with what rob just said the the name of the game here is guessing the power

226
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right you're given a base we know so we know a base to a power and we know the answer

227
00:15:57,040 --> 00:16:06,400
right the discrete law we know that we we we know the answer the answer is um the public key which

228
00:16:06,400 --> 00:16:13,200
is your you know your private key times a generator which means it's a generate you you've taken the

229
00:16:13,200 --> 00:16:19,820
generator point that everybody knows and we've added it to itself a number of times right so it's

230
00:16:19,820 --> 00:16:28,100
that number of times is we call it scalar k times g sorry um your public you're sorry your private

231
00:16:28,100 --> 00:16:32,020
key times the generator is the amount of times we've added it to each other and we tell the world

232
00:16:32,020 --> 00:16:38,120
this number and we're so confident and brazen that we're saying guess what we can tell you this total

233
00:16:38,120 --> 00:16:44,520
number you'll never know you'll never be able to know how many times i i added my public key together

234
00:16:44,520 --> 00:16:51,060
Sorry, I added – you'll never know how many times I added the generator together to exponentiate into the –

235
00:16:51,060 --> 00:16:54,340
And that number of times, that scalar is your private key.

236
00:16:54,780 --> 00:16:55,840
Yep, and that's right.

237
00:16:56,020 --> 00:16:56,960
So come and get it.

238
00:16:57,180 --> 00:17:01,420
You're basically – the discrete logarithm problem says come and fucking take it, right?

239
00:17:01,600 --> 00:17:01,780
Yeah.

240
00:17:01,980 --> 00:17:06,500
I'm giving you all of the pieces and I'm so confident in this math.

241
00:17:06,500 --> 00:17:10,100
I'm confident in the number of computing,

242
00:17:10,580 --> 00:17:13,660
the amount of computing resources that exists today

243
00:17:13,660 --> 00:17:15,540
and reasonably into the future

244
00:17:15,540 --> 00:17:18,160
that I'm telling you to come and take it.

245
00:17:18,780 --> 00:17:21,560
Now, I'm going to repeat myself a few times

246
00:17:21,560 --> 00:17:24,740
because I'm trying to zero in on the factors

247
00:17:24,740 --> 00:17:29,160
because I think it's important first

248
00:17:29,160 --> 00:17:32,540
to explain what a logarithm is to a lot of people.

249
00:17:34,000 --> 00:17:35,620
I don't know about you, Rob,

250
00:17:35,620 --> 00:17:41,840
But when I was in high school, which I barely I really barely went to my second and third year, I kind of stopped going.

251
00:17:41,920 --> 00:17:44,140
But I was still into math and I was studying it by myself.

252
00:17:44,500 --> 00:17:47,700
But when I hit this thing called logarithm, my brain broke.

253
00:17:49,760 --> 00:17:52,180
And I actually gave up.

254
00:17:52,220 --> 00:17:52,840
I said, you know what?

255
00:17:52,840 --> 00:17:54,820
I guess I'm never going to understand what a logarithm is.

256
00:17:54,840 --> 00:17:58,440
And then in college, I learned calculus and it all made sense.

257
00:17:58,440 --> 00:17:59,820
And I was like, oh, my God.

258
00:18:00,740 --> 00:18:03,000
That was like one of the first moments I got angry.

259
00:18:03,100 --> 00:18:04,800
I'm like, why don't they teach calculus first?

260
00:18:04,800 --> 00:18:05,820
Why are they doing this?

261
00:18:05,880 --> 00:18:07,040
Why do they do it like this?

262
00:18:07,320 --> 00:18:13,140
You know, this is absurd because I remembered all of the pain I went through in high school

263
00:18:13,140 --> 00:18:16,040
trying to because I'm like, I've never had this problem before.

264
00:18:16,100 --> 00:18:16,900
I've never had anything.

265
00:18:17,000 --> 00:18:18,080
I just like had to accept.

266
00:18:18,360 --> 00:18:19,460
I wasn't going to get.

267
00:18:20,120 --> 00:18:22,740
But why did this word is weird?

268
00:18:23,640 --> 00:18:25,060
Logarithm, it's a weird word.

269
00:18:25,060 --> 00:18:30,460
And it didn't for some reason, my brain, it just could not it just could not work.

270
00:18:30,460 --> 00:18:35,920
and no matter how many times I sat there and did problems about it.

271
00:18:36,080 --> 00:18:38,660
So let me – I just got done explaining this to my daughter

272
00:18:38,660 --> 00:18:40,940
and I feel – I'm feeling like I'm on a heat check.

273
00:18:41,340 --> 00:18:41,880
Go for it.

274
00:18:42,120 --> 00:18:44,580
I'm going to talk about what a logarithm is, OK?

275
00:18:45,440 --> 00:18:47,500
Because I don't want people to get tripped up by this

276
00:18:47,500 --> 00:18:50,060
and you can't know what a discrete logarithm is

277
00:18:50,060 --> 00:18:51,740
if you don't know what a logarithm is, OK?

278
00:18:53,080 --> 00:18:57,000
So all the logarithm is is a power, OK?

279
00:18:57,000 --> 00:19:05,720
I mean, that's the first thing that I know this seems this is going to seem stupid to a few people here, but like all logarithm is is a power.

280
00:19:05,880 --> 00:19:14,940
So in other words, if I want to find a logarithm of we talk in shorthand, this is part of the problem because there's sort of an infinite amount of ways to express a logarithm.

281
00:19:15,320 --> 00:19:20,220
But, you know, one of the shorthand ways we like is base 10 because that's how we think.

282
00:19:20,220 --> 00:19:28,080
Right. So if the base 10 logarithm is the 10 to the X equals some number.

283
00:19:28,200 --> 00:19:30,480
Right. We want to know what X is.

284
00:19:30,780 --> 00:19:32,880
Now, a lot of these we can do in our head.

285
00:19:33,280 --> 00:19:36,500
Like the base 10 logarithm of 10 is one.

286
00:19:36,500 --> 00:19:39,120
The base 10 logarithm of 100 is two.

287
00:19:39,600 --> 00:19:41,360
Right. 10 times 10 is 100.

288
00:19:41,840 --> 00:19:44,720
You just count. That's right. You just count the zeros.

289
00:19:45,300 --> 00:19:47,340
You count the zeros, buddy, and you are done.

290
00:19:47,340 --> 00:19:55,140
right base 10 logarithm of 10 to the x 10 to the c or 10 to any number right is that number

291
00:19:55,140 --> 00:20:03,080
right um i don't think i've gotten into toilet seat math yet on this podcast not yet that's like

292
00:20:03,080 --> 00:20:08,800
one of my big secrets of how i taught my oldest daughter yes toilet seat math i was good and it

293
00:20:08,800 --> 00:20:15,060
really out it was out of like absolute frustration of trying to explain like how do you not know that

294
00:20:15,060 --> 00:20:22,640
Like I was trying to explain like terms to my oldest daughter who's now like a math major and has exceeded me.

295
00:20:23,360 --> 00:20:25,020
You'll hear her on this podcast eventually.

296
00:20:25,440 --> 00:20:42,713
But I was like OK if I have 2x plus 9x how much do I have And she like I don know I don know what that means I like well if you have two I got so frustrated I like all right if you have two toilet seats and you add nine toilet seats to that what do you have And she like oh well I got 11 toilet seats I like okay

297
00:20:42,773 --> 00:20:47,933
well, you know, what if toilet seat was just an avatar for anything, right? Is your variable.

298
00:20:48,453 --> 00:20:53,253
And so I was able to use, I like my standard notation for variables for teaching things like

299
00:20:53,253 --> 00:21:00,173
that is toilet seats, doodos, and like things that are clearly no one would ever use that

300
00:21:00,173 --> 00:21:03,433
are absurd enough that a child actually kind of knows now.

301
00:21:03,513 --> 00:21:04,533
Oh, it could be anything.

302
00:21:05,593 --> 00:21:07,833
It could be anything, right?

303
00:21:08,093 --> 00:21:08,753
Yes, right.

304
00:21:09,373 --> 00:21:15,493
And so I was teaching the thing that resonated with my younger daughter who kind of really

305
00:21:15,493 --> 00:21:20,113
she struggles to get like she struggles to get good grades in math, even though she kind

306
00:21:20,113 --> 00:21:21,013
of gets it.

307
00:21:21,013 --> 00:21:23,013
So she's a lot like me that way.

308
00:21:23,253 --> 00:21:30,173
I was so we got to the point where we said the base is the toilet the base toilet seat

309
00:21:30,173 --> 00:21:42,293
log right of I should say the base 10 log of 10 to the toilet seat is the toilet seat I can't

310
00:21:42,293 --> 00:21:48,513
believe I'm doing this on this podcast right now but like you know and if you don't like x this is

311
00:21:48,513 --> 00:21:51,073
I guess because, you know, that wasn't for me, it wasn't enough.

312
00:21:51,413 --> 00:22:00,453
X wasn't enough to, you know, for some people are mature enough that they can use X, Y and Z and do learn algebra.

313
00:22:00,633 --> 00:22:01,853
For me, that was never the case.

314
00:22:01,913 --> 00:22:02,793
I needed to go further.

315
00:22:02,913 --> 00:22:05,233
I need a more absurd example.

316
00:22:05,913 --> 00:22:11,773
So but this is all to say that this notion of a logarithm is nothing more than a power.

317
00:22:11,773 --> 00:22:20,833
And so I think maybe it's useful really quick to take one more step down and to say, do we know what PEMDAS is?

318
00:22:21,093 --> 00:22:22,653
Do we understand PEMDAS?

319
00:22:23,293 --> 00:22:24,553
Please excuse my dear Aunt Sally.

320
00:22:25,033 --> 00:22:25,653
Please, yes.

321
00:22:25,733 --> 00:22:31,733
For our generation, and you're 50 years younger than me, but still for our generation.

322
00:22:32,293 --> 00:22:33,233
There's a generation.

323
00:22:33,533 --> 00:22:37,493
I think if you study, there's a generation that got, please excuse my dear Aunt Sally.

324
00:22:37,493 --> 00:22:42,213
but that became way too long for the gen alphas.

325
00:22:43,173 --> 00:22:45,213
So now they have PEMDAS.

326
00:22:45,453 --> 00:22:48,433
It's the same acronym, P-E-M-D-A-S,

327
00:22:48,633 --> 00:22:49,393
and it's just an order.

328
00:22:49,653 --> 00:22:52,713
This gives you the order of operations

329
00:22:52,713 --> 00:22:54,373
of what you do first in algebra.

330
00:22:54,993 --> 00:22:58,613
Starts with parentheses, then exponents, right?

331
00:22:58,973 --> 00:23:01,573
Your brain should be going off right now for saying,

332
00:23:01,673 --> 00:23:04,153
oh yeah, he's talking about something about exponents

333
00:23:04,153 --> 00:23:07,713
because that's what we're talking about with logarithms.

334
00:23:07,793 --> 00:23:09,453
So we've got parentheses, exponents,

335
00:23:10,053 --> 00:23:11,733
and we've got multiplication and division,

336
00:23:11,733 --> 00:23:12,873
which sit on their own,

337
00:23:13,393 --> 00:23:15,433
like those sit on their own level, right?

338
00:23:15,813 --> 00:23:16,573
Multiplication and division,

339
00:23:16,673 --> 00:23:17,893
it doesn't matter which one you do first.

340
00:23:18,053 --> 00:23:19,013
It just matters that...

341
00:23:19,013 --> 00:23:19,793
Effectively the same thing.

342
00:23:20,193 --> 00:23:21,153
Effectively the same thing.

343
00:23:21,213 --> 00:23:23,613
And then you have addition and subtraction,

344
00:23:23,673 --> 00:23:24,433
the A and the S,

345
00:23:24,493 --> 00:23:26,253
that's also effectively the same thing.

346
00:23:26,313 --> 00:23:28,233
As long as you know that multiplication and division

347
00:23:28,233 --> 00:23:31,473
is before addition and subtraction, you're good.

348
00:23:31,573 --> 00:23:33,113
As long as you know that exponentiation

349
00:23:33,113 --> 00:23:38,353
is before multiplication slash division, you're good.

350
00:23:38,353 --> 00:23:45,193
But unfortunately, the acronym and the fun saying

351
00:23:45,193 --> 00:23:52,833
doesn't pair exponentiation with its partner called the logarithm.

352
00:23:53,793 --> 00:23:54,093
That's right.

353
00:23:54,593 --> 00:24:00,113
And so if we just kind of go back and help help that,

354
00:24:00,233 --> 00:24:02,133
it's just not it's not great.

355
00:24:02,133 --> 00:24:08,853
please excuse, you know, lovely, there's no good, there was, I'm pretty sure people tried to do this

356
00:24:08,853 --> 00:24:14,313
and it never worked out. So we just have to fill this gap in right now and say, if you go and think

357
00:24:14,313 --> 00:24:19,653
of PEMDAS and if you think of associate, if you think of addition and subtraction as similar,

358
00:24:20,153 --> 00:24:26,653
right, as buddies, twins, little twinsies, right. And then you think of multiplication division as

359
00:24:26,653 --> 00:24:32,393
older twins, right? Imagine the horrors of this parent just continuing to have twins,

360
00:24:32,393 --> 00:24:37,013
but like, and then you go back and then you go up the level, the big brother, the big sister,

361
00:24:37,013 --> 00:24:44,813
you have exponentiation and logarithms. Okay. So they're, they're twins, you know, and you know

362
00:24:44,813 --> 00:24:49,793
what's great about twins? They undo each other. They love to undermine each other, right? So

363
00:24:49,793 --> 00:24:51,993
So subtraction undermines addition.

364
00:24:52,553 --> 00:24:54,293
Division undermines multiplication.

365
00:24:55,253 --> 00:24:59,673
And logarithms undermine exponentiation.

366
00:25:00,113 --> 00:25:01,273
They undo it.

367
00:25:03,093 --> 00:25:06,833
I'm hoping this enriches this discussion.

368
00:25:07,553 --> 00:25:08,633
So what is a logarithm?

369
00:25:08,753 --> 00:25:11,953
So every next step, by the way,

370
00:25:12,473 --> 00:25:17,813
multiplication is nothing more than repeated addition, right?

371
00:25:18,513 --> 00:25:18,693
Yeah.

372
00:25:18,693 --> 00:25:21,853
So multiplication is – let's repeat that.

373
00:25:22,533 --> 00:25:23,953
Multiplication is repeated addition.

374
00:25:24,093 --> 00:25:31,193
You take addition, you multiply it by a scalar, and that's just how you repeat addition, just like we just said happens on an elliptic curve.

375
00:25:31,513 --> 00:25:43,513
You take your generator point, and you – we haven't gotten here yet, but spoiler alert, you're going to do an operation called addition repeatedly, right?

376
00:25:43,713 --> 00:25:44,833
How many times?

377
00:25:45,413 --> 00:25:47,873
Well, I know, but you don't, right?

378
00:25:48,693 --> 00:25:49,913
Certain amount of times.

379
00:25:50,413 --> 00:25:56,493
And I'm actually, you know, I'm so confident that you can't even guess how many times because it's hard to do.

380
00:25:56,813 --> 00:25:57,373
It's just hard.

381
00:25:57,873 --> 00:25:58,113
Okay.

382
00:25:58,633 --> 00:26:01,673
So multiplication is repeated addition.

383
00:26:02,873 --> 00:26:05,713
Exponentiation is repeated multiplication.

384
00:26:06,493 --> 00:26:07,153
Okay, right?

385
00:26:07,213 --> 00:26:08,733
So I'm really just drawing this line.

386
00:26:08,833 --> 00:26:12,613
And so there's no special magic or sorcery here.

387
00:26:12,613 --> 00:26:19,833
This is just another logical step in how we do arithmetic, right?

388
00:26:20,913 --> 00:26:23,493
Now, what's repeated exponentiation?

389
00:26:23,813 --> 00:26:32,253
I don't want to introduce that because it's not relevant and we need to just stick to what – we need to just try to rock this here, right?

390
00:26:32,813 --> 00:26:40,833
So we have, once again, very, very base substrate addition that gets undermined by subtraction, okay?

391
00:26:40,833 --> 00:26:46,573
Then we have repeated addition called multiplication, which can get undone by division.

392
00:26:46,853 --> 00:26:48,073
I think we all get this, right?

393
00:26:48,453 --> 00:26:53,853
And so it's one little extra step to say, well, repeated multiplication is exponentiation.

394
00:26:54,973 --> 00:26:56,553
And that gets undone.

395
00:26:57,713 --> 00:27:04,173
And when I say undone, remember the conversation about groups where we said every group has an identity element?

396
00:27:05,173 --> 00:27:07,693
By undone, it means we do the operation.

397
00:27:07,693 --> 00:27:12,293
it's the thing we do to get the identity as the answer.

398
00:27:13,993 --> 00:27:14,673
Right?

399
00:27:15,413 --> 00:27:19,253
So, and addition has the identity of zero.

400
00:27:20,733 --> 00:27:23,173
Multiplication has an identity of one.

401
00:27:24,213 --> 00:27:26,973
The question now is what's the identity,

402
00:27:27,473 --> 00:27:30,233
and boy, this is now tightrope walk

403
00:27:30,233 --> 00:27:32,013
because I have not thought through this at all.

404
00:27:32,333 --> 00:27:36,013
But is there an identity element for exponentiation?

405
00:27:36,473 --> 00:27:37,513
I think there is.

406
00:27:37,693 --> 00:27:38,913
I think there is too.

407
00:27:39,633 --> 00:27:44,993
And I think it's the base that you're raising to a power.

408
00:27:48,833 --> 00:27:49,313
Right?

409
00:27:51,493 --> 00:27:56,713
So in other words, if I said, what's the, what's the, that, that, and that's why the base matters, right?

410
00:27:56,713 --> 00:28:02,313
If I said, what's the base 10 log of a thousand, right?

411
00:28:02,313 --> 00:28:10,033
Then it's the question is what – the logarithm is the thing that would get me back to 10, right?

412
00:28:10,333 --> 00:28:14,933
So if I take 1,000 and it's the somethingth root, right?

413
00:28:14,993 --> 00:28:17,293
It's the opposite of exponentiation.

414
00:28:17,613 --> 00:28:21,533
The opposite of exponentiation is the root, right?

415
00:28:22,053 --> 00:28:23,253
We know square roots.

416
00:28:23,553 --> 00:28:25,233
Those are nice, right?

417
00:28:25,573 --> 00:28:31,573
If you want to take a perfect square and undo it into its root, say, oh, it's 49.

418
00:28:31,573 --> 00:28:32,993
Well, there's two answers.

419
00:28:33,133 --> 00:28:35,013
It's plus or minus seven, right?

420
00:28:36,053 --> 00:28:37,713
Any one of those two things.

421
00:28:38,093 --> 00:28:44,413
But if I'm really trying to get to the base, so base 10, the answer's of 1,000, the answer's three.

422
00:28:44,673 --> 00:28:49,973
If I want to get the base two logarithm of the number eight, the answer's three.

423
00:28:50,713 --> 00:28:53,953
I'm hoping that was pretty, I'm hoping that caught, right?

424
00:28:54,433 --> 00:28:57,313
Because two to the third is eight, right?

425
00:28:57,553 --> 00:29:00,573
So I'm asking the question, what's the power?

426
00:29:00,573 --> 00:29:08,613
right what's the power that does this that does it and that and the reason we asked that question

427
00:29:08,613 --> 00:29:16,873
right is because that is a very hard thing to do hard enough to protect

428
00:29:16,873 --> 00:29:24,453
as much wealth as i'm willing to protect underneath it for we'll just say

429
00:29:24,453 --> 00:29:28,313
for a reasonable time on earth.

430
00:29:28,673 --> 00:29:29,133
That's right.

431
00:29:32,193 --> 00:29:33,573
I'll stop there.

432
00:29:34,793 --> 00:29:37,313
No, I think it's a good pause point.

433
00:29:40,713 --> 00:29:42,153
Maybe to even tie back now,

434
00:29:42,193 --> 00:29:43,093
we're talking about logs,

435
00:29:44,233 --> 00:29:45,493
kind of the asymmetry of them,

436
00:29:45,853 --> 00:29:49,573
specifically the discrete log problem

437
00:29:49,573 --> 00:29:53,313
and what would be the...

438
00:29:54,453 --> 00:30:09,713
Did I actually meet myself?

439
00:30:10,033 --> 00:30:10,473
You did.

440
00:30:13,473 --> 00:30:17,693
You said back to the discrete log problem and then we lost you.

441
00:30:22,233 --> 00:30:23,193
All that wisdom.

442
00:30:23,193 --> 00:30:29,853
all that wisdom is gone forever now yeah that's correct so going through logarithms going through

443
00:30:29,853 --> 00:30:36,193
kind of like the asymmetry this really large search space of two to the 256 like what makes it

444
00:30:36,193 --> 00:30:41,373
the word discrete log problem we're talking about logarithms in general and how that's the opposite

445
00:30:41,373 --> 00:30:50,773
almost the inverse of expense expansion expensation you got it and so this is one of my favorite

446
00:30:50,773 --> 00:30:55,373
topics ever, by the way, is the fact that it's called a discrete log problem because

447
00:30:55,373 --> 00:30:56,953
it has double meaning.

448
00:30:58,493 --> 00:30:59,013
Yes.

449
00:31:00,733 --> 00:31:10,173
And so for a discrete log, I guess what you have to do, you're not able to reasonably

450
00:31:10,173 --> 00:31:13,413
ascertain at any point if you're closer or not to the final answer.

451
00:31:13,953 --> 00:31:15,433
It's like a continuous function.

452
00:31:15,433 --> 00:31:23,913
So it's kind of like a total random space where you're – and this is actually just as a very brief aside and interesting.

453
00:31:24,573 --> 00:31:31,553
I mentioned earlier Bitcoin mining and like proof of work and this asymmetry that's trivial to verify.

454
00:31:31,873 --> 00:31:42,053
The difference between proof of work, though, and a public-private key pair is that there's only one real answer for the private key lookup.

455
00:31:42,053 --> 00:31:50,613
Whereas in Bitcoin mining, there's a difficulty target where there's actually many possible solutions to the next valid Bitcoin block.

456
00:31:51,313 --> 00:31:54,633
For public-private key, there's only an answer.

457
00:31:55,253 --> 00:32:02,333
And so with Bitcoin mining, when you're randomly hashing, you're kind of going along, you are just pulling lotto tickets to find the one with the most leading zeros.

458
00:32:02,733 --> 00:32:07,253
But it's not like the search spaces, there's only one number that satisfies that.

459
00:32:07,253 --> 00:32:13,213
And that's also part of the unpredictable nature of how often in frequency blocks come in.

460
00:32:13,253 --> 00:32:14,533
They come in at a Poisson distribution.

461
00:32:14,673 --> 00:32:15,913
Maybe we should talk about that another time.

462
00:32:17,373 --> 00:32:29,533
But the idea here, though, is that from each incremental step of trying to undo the public-private key pair, since the function is continuous, there isn't really marginal more information that you're gathering along the way.

463
00:32:32,173 --> 00:32:35,033
Yeah, I mean this is a rich topic in and of itself.

464
00:32:35,033 --> 00:32:42,153
But like I think I've talked to you a little bit about this just on the side, trying to really grok it.

465
00:32:42,293 --> 00:32:50,913
But something that I think is important that needs to just – we need to just kind of sit with in our journey here.

466
00:32:51,593 --> 00:32:56,533
One of the big distinctions in math is whether something is continuous or whether it's discrete.

467
00:32:56,533 --> 00:33:11,853
Right. When I say something, I should probably be specific, whether like a sequence of events or whether a set, right, a set of, you know, a set of possible numbers.

468
00:33:12,853 --> 00:33:18,513
Whether a set of possible numbers is continuous or discrete and to try to really kind of drive it home.

469
00:33:18,513 --> 00:33:33,553
Just imagine like a circle, right? A circle is clearly a continuous, you know, it's a continuous object, meaning you can draw it without lifting your pencil, right?

470
00:33:33,753 --> 00:33:41,633
But it's more than a, it's actually more than a continuous object. It's a very specific type of continuous object.

471
00:33:41,633 --> 00:33:48,293
like a square is a continuous object except we have we know how many sides are on a square right

472
00:33:48,293 --> 00:33:53,693
we know we can see the four sides of a square right right we can see the five sides of a pentagon

473
00:33:53,693 --> 00:34:00,493
we can see the 10 sides of a decagon right a circle is when the number of sides

474
00:34:00,493 --> 00:34:09,473
tends to infinity and you can't and then the number of then the number you the zero it's like

475
00:34:09,473 --> 00:34:11,753
There are no sides, right?

476
00:34:11,933 --> 00:34:22,933
So it's like a special type of continuous where it's actually convergent, sort of converging on infinite, right?

477
00:34:23,233 --> 00:34:29,213
And that's when, you know, if we think about time and how we observe trials in time, right?

478
00:34:29,233 --> 00:34:31,413
We observe them discreetly, right?

479
00:34:31,493 --> 00:34:32,833
Then we can see the points.

480
00:34:33,993 --> 00:34:35,893
Market data is like that, right?

481
00:34:35,893 --> 00:34:37,313
You can see every tick.

482
00:34:39,473 --> 00:34:43,453
Or you can say, hey, every millisecond, show me the price of something every millisecond.

483
00:34:43,453 --> 00:34:53,253
But you can't say, show me the price of something continuously, even though we all know that time is continuous.

484
00:34:53,593 --> 00:34:55,133
We all know that it's all happening.

485
00:34:55,133 --> 00:35:00,553
It happens continuously, but it can only be observed discreetly.

486
00:35:01,913 --> 00:35:06,553
Okay, so this is like the just want to kind of like, this is a dichotomy that's really important.

487
00:35:06,553 --> 00:35:13,373
and it happens to be important when we talk about the elliptic curve of bitcoin this is so this is

488
00:35:13,373 --> 00:35:19,693
why we're spending time on it because it happens to be an important thing um directionality is an

489
00:35:19,693 --> 00:35:25,953
important thing as well and when you know i was trained in probability that the object of the game

490
00:35:25,953 --> 00:35:31,013
was to take these discrete trials and to be able to view them continuously and so everyone knows

491
00:35:31,013 --> 00:35:36,693
what the law of large numbers is. That's the thing that says, you know, your discrete distribution,

492
00:35:36,693 --> 00:35:41,493
if you do it long enough, you can just assume it's continuous. And then you can, then what you

493
00:35:41,493 --> 00:35:47,913
can do is the math gets more elegant. That's, I guess it gets more elegant and easy. And when you

494
00:35:47,913 --> 00:35:52,553
are a human being doing math, that's what you want. But when you're trying to protect,

495
00:35:52,553 --> 00:35:59,373
when you're trying to protect mathematical objects, right,

496
00:35:59,693 --> 00:36:01,393
behind that work, you want the opposite.

497
00:36:01,753 --> 00:36:03,573
So you want to take something that's continuous

498
00:36:03,573 --> 00:36:05,633
and if you can, express it discreetly

499
00:36:05,633 --> 00:36:07,973
because then you're going to make it very ugly

500
00:36:07,973 --> 00:36:10,233
and certainly a human being can't do it

501
00:36:10,233 --> 00:36:14,413
and it turns out you want to make it so hard that computers can't do it.

502
00:36:14,413 --> 00:36:16,833
This goes to the entire field of cryptography

503
00:36:16,833 --> 00:36:21,613
is trying to at every point basically shake the Etch-A-Sketch board,

504
00:36:21,613 --> 00:36:27,333
make it a discrete problem not a continuous one if it were a continuous problem you'd be able to

505
00:36:27,333 --> 00:36:33,393
reasonably discern oh am i closer not to guessing the private key right i'd be like oh maybe i just

506
00:36:33,393 --> 00:36:38,133
need to increase the number by one or two right like you would know you would have like a general

507
00:36:38,133 --> 00:36:45,493
like algorithm that you could recursively run to quickly get to the answer and this goes back to

508
00:36:45,493 --> 00:36:58,153
It's just the, while we have this curve of the elliptic curve, you don't just go, it's not like the private key is going to be next to the public key for every single point.

509
00:36:58,153 --> 00:37:01,213
It's randomly somewhere across that entire curve.

510
00:37:02,573 --> 00:37:12,113
And because like each number is like a dot, as opposed, instead of viewing it as the continuous graph view, each number is a point along that curve.

511
00:37:12,113 --> 00:37:15,873
and each incremental step through the generator point

512
00:37:15,873 --> 00:37:18,353
throws you out somewhere entirely random.

513
00:37:19,413 --> 00:37:22,113
And that happens again and again and again

514
00:37:22,113 --> 00:37:26,593
for the amount of times you pick a point.

515
00:37:28,033 --> 00:37:30,013
It'll be constantly randomly shuffling around

516
00:37:30,013 --> 00:37:32,473
until you get to the final number of points in your generator,

517
00:37:33,073 --> 00:37:34,513
which is your private key.

518
00:37:37,513 --> 00:37:41,113
Yeah, and it's meant to be hard and ugly

519
00:37:41,113 --> 00:37:50,253
and um on purpose and it and every like trial that you try doesn't give you any additional

520
00:37:50,253 --> 00:38:01,073
information about where you are in this process right yeah yeah so um it turns out and so here's

521
00:38:01,073 --> 00:38:08,213
the other thing if you guys if you ever really sat and spent time doing arithmetic right um and

522
00:38:08,213 --> 00:38:13,293
you worked with powers, you would, even with calculators, you kind of like, oh my God, why

523
00:38:13,293 --> 00:38:17,173
is my cat? Well, I have, I have a lot of function to my calculator and they work instantly except

524
00:38:17,173 --> 00:38:22,533
for one, which is like solve the power. And it's not like a lot, like, you know, your power,

525
00:38:22,633 --> 00:38:26,213
your calculator probably is a base 10 log and a natural log that does that very fast.

526
00:38:26,773 --> 00:38:32,693
But my calculator, my HP 12 C is a financial calculator. And a lot of times you want to know,

527
00:38:32,693 --> 00:38:36,873
like you have the, you have a compound interest rate, like one plus I to a certain power. And

528
00:38:36,873 --> 00:38:40,393
You want to solve for the power that gives you like you have an annuity.

529
00:38:40,513 --> 00:38:42,693
You have a stream of payments for the next 50 years.

530
00:38:42,693 --> 00:38:50,593
And you say, well, find me the N that is actually, you know, that solves the problem.

531
00:38:50,593 --> 00:38:53,033
And you notice the calculator takes a long time.

532
00:38:53,933 --> 00:38:57,813
So this is something that happened at a fairly young age for me, right?

533
00:38:57,853 --> 00:39:06,733
It's like, oh, it takes quite a long time to calculate a power, even a calculator that does everything else at just absolute rocket speeds, right?

534
00:39:06,873 --> 00:39:14,453
so we start to ask these questions like what is it that makes this problem hard right solving

535
00:39:14,453 --> 00:39:21,753
finding an exponent and discreteness right discreteness is actually part of what makes it hard

536
00:39:21,753 --> 00:39:26,973
um i want to try to like really nail like land this point i don't feel like i have but like

537
00:39:26,973 --> 00:39:32,053
what we call analytical functions you know so like if you had a circle x squared plus y squared

538
00:39:32,053 --> 00:39:41,513
equals one and you were given like um some you were given something some some answer that tells

539
00:39:41,513 --> 00:39:47,053
you something about the points on that curve like it probably wouldn't be hard to just solve for

540
00:39:47,053 --> 00:39:52,133
you can solve the equation literally solve the equation for what x and y are without having to

541
00:39:52,133 --> 00:39:58,713
brute force and you know what discreetness does is it makes you have it make it turns it into a

542
00:39:58,713 --> 00:39:59,673
trial and error problem.

543
00:40:01,293 --> 00:40:02,733
It's a problem you can't

544
00:40:02,733 --> 00:40:04,613
solve algebraically and you have to do

545
00:40:04,613 --> 00:40:06,213
trial and error.

546
00:40:06,933 --> 00:40:07,153
And so

547
00:40:07,153 --> 00:40:10,453
removing the continuity

548
00:40:10,453 --> 00:40:12,333
and reducing it to discreteness is

549
00:40:12,333 --> 00:40:14,653
really one of the things that

550
00:40:14,653 --> 00:40:16,033
actually does make it hard.

551
00:40:17,573 --> 00:40:18,713
You're still on mute

552
00:40:18,713 --> 00:40:19,293
again.

553
00:40:20,213 --> 00:40:22,653
Sorry, I'm trying to be a good

554
00:40:22,653 --> 00:40:24,433
audio engineer. I need a hand signal.

555
00:40:24,613 --> 00:40:25,173
I need a hand signal.

556
00:40:26,253 --> 00:40:28,513
This is an interesting thing too in that

557
00:40:28,513 --> 00:40:34,453
the battle-testedness of Bitcoin's discrete log problem

558
00:40:34,453 --> 00:40:36,153
is something that is inferred.

559
00:40:36,413 --> 00:40:38,913
No one's written a mathematical proof

560
00:40:38,913 --> 00:40:44,473
of being able to show that it's P equal NP hard.

561
00:40:48,133 --> 00:40:51,653
And it's just something that very publicly in the open for decades

562
00:40:51,653 --> 00:40:53,933
people have tried to break this and no one's been able to.

563
00:40:54,533 --> 00:40:55,733
Proof by come and take it.

564
00:40:56,673 --> 00:40:56,913
Yeah.

565
00:40:56,913 --> 00:40:59,553
it's pure applied

566
00:40:59,553 --> 00:41:00,713
pragmatic

567
00:41:00,713 --> 00:41:22,506
we now have a multi dollar asset running this let alone decades before that people were using this cryptography So the ultimate what been able to see at this point is that the best people have been able to do is the square root of N as we talked about earlier which is not cutting it in half

568
00:41:22,506 --> 00:41:23,986
It's taking the square root, right?

569
00:41:23,986 --> 00:41:43,446
So it's, you know, if you're talking about expensiation and, you know, taking the square root as a form of an exponent, in a way, to the one-half power, it still is nowhere near good enough to be able to make something that's practically insolvable or reversible in our lifetimes.

570
00:41:43,446 --> 00:41:47,646
This as a – we don't have to go deep into it, but just a little sprinkle.

571
00:41:48,226 --> 00:41:59,846
This is where the quantum stuff starts breaking assumptions is the idea that you could actually begin to infer information rather than having no information at each incremental stuff.

572
00:42:01,426 --> 00:42:09,246
Let me give a quick example of why the square root thing actually halves the solution space.

573
00:42:09,246 --> 00:42:17,226
you know um just if you wanted to know what the square root of 100 is and you you know you're five

574
00:42:17,226 --> 00:42:22,106
years old you haven't been taught the square root of 100 is 10 so you start plugging numbers in on

575
00:42:22,106 --> 00:42:26,086
your calculator what's the square root of 100 you might you know you might to yourself think man i

576
00:42:26,086 --> 00:42:31,646
have to take i have to try every number right i'm going to do one squared two squared three squared

577
00:42:31,646 --> 00:42:37,126
four squared it's just that the thing is after you hit 10 and you start going above 100

578
00:42:37,126 --> 00:42:45,266
you start knowing, okay, well, okay, clearly, okay, I have to go up to 10.

579
00:42:45,266 --> 00:42:49,646
Then you might, there's a property where you might, where you realize, okay,

580
00:42:49,866 --> 00:42:53,926
I might've stepped in shit here because I'm not ready to land this plane.

581
00:42:55,746 --> 00:42:57,266
Just go for it.

582
00:43:00,786 --> 00:43:05,706
At some point, I actually, I'm kind of stuck.

583
00:43:07,126 --> 00:43:08,266
I really screwed up.

584
00:43:08,326 --> 00:43:09,046
I didn't think this through.

585
00:43:13,306 --> 00:43:15,386
I'm thinking of the sieve of Aristocanese,

586
00:43:15,946 --> 00:43:19,406
if you're familiar with that in terms of how to, like,

587
00:43:20,366 --> 00:43:21,906
say, find all the prime numbers.

588
00:43:22,846 --> 00:43:27,146
Say you want to find all the prime numbers between 0 and 100, right?

589
00:43:28,566 --> 00:43:33,426
You really, you know, I think I'm just making it worse.

590
00:43:33,426 --> 00:43:39,626
i really want i'm going to pull the ripcord here and tap out i'm sorry this is going to happen

591
00:43:39,626 --> 00:43:43,386
episode six this is like the first time it happens

592
00:43:43,386 --> 00:43:51,866
you're on you're on mute god damn it i need you here man i'm listening i'm listening um

593
00:43:51,866 --> 00:43:56,646
the steve arisosthenes is you know way of being able to find prime numbers by going through all

594
00:43:56,646 --> 00:44:01,746
the competent metrics uh up to a given limit it's like a process of elimination way of being

595
00:44:01,746 --> 00:44:08,026
able to do primes and what i would say is essentially what the idea of what a quantum

596
00:44:08,026 --> 00:44:18,706
computer could do is that because it's basically further able to the idea at the highest level i'm

597
00:44:18,706 --> 00:44:25,166
not a quantum expert but it's the idea that it actually can parallelize and actually further

598
00:44:25,166 --> 00:44:26,946
recursively undo

599
00:44:26,946 --> 00:44:28,646
the

600
00:44:28,646 --> 00:44:31,246
expense dentiation you do by using

601
00:44:31,246 --> 00:44:33,326
the generator point in a way

602
00:44:33,326 --> 00:44:35,326
that it's not going to

603
00:44:35,326 --> 00:44:37,186
linearly, internally

604
00:44:37,186 --> 00:44:39,326
in binary space, like, check all

605
00:44:39,326 --> 00:44:41,286
the guesses and be able to further get to

606
00:44:41,286 --> 00:44:42,046
the final answer.

607
00:44:45,566 --> 00:44:46,666
It's something that

608
00:44:46,666 --> 00:44:49,126
in the quantum space at a high level,

609
00:44:50,126 --> 00:44:51,006
I mean,

610
00:44:51,166 --> 00:44:53,186
when it comes up in Bitcoin, I actually sit

611
00:44:53,186 --> 00:44:55,146
in a space where I think that there should be time and effort

612
00:44:55,146 --> 00:44:57,266
put into this stuff and thinking about what this looks like.

613
00:44:57,286 --> 00:45:00,946
If this is an intergenerational project and in the next 10 years,

614
00:45:00,946 --> 00:45:02,126
there's meaningful breakthroughs,

615
00:45:02,126 --> 00:45:04,406
I think it's important for Bitcoin to be well ahead of that.

616
00:45:06,126 --> 00:45:08,986
But you have to figure out a new way of doing a signing algorithm because you

617
00:45:08,986 --> 00:45:11,806
basically have to throw out all of the elliptic curve cryptography into the

618
00:45:11,806 --> 00:45:12,186
garbage.

619
00:45:12,926 --> 00:45:16,046
And this goes into something we mentioned previously that quantum computers are

620
00:45:16,046 --> 00:45:20,986
inherently better at attacking public private key cryptography than they are,

621
00:45:21,086 --> 00:45:22,606
say, dealing with a hash function.

622
00:45:22,606 --> 00:45:25,746
it's a totally different algorithm

623
00:45:25,746 --> 00:45:27,626
which I'll just call it

624
00:45:27,626 --> 00:45:29,726
Shor's algorithm is the ability

625
00:45:29,726 --> 00:45:31,746
to kind of like accelerate

626
00:45:31,746 --> 00:45:33,546
the computation for public private key pairs

627
00:45:33,546 --> 00:45:34,566
hypothetically

628
00:45:34,566 --> 00:45:37,666
on computer and Grover's algorithm

629
00:45:37,666 --> 00:45:39,366
is being able to reverse engineer

630
00:45:39,366 --> 00:45:40,566
a hash and a pre-image

631
00:45:40,566 --> 00:45:43,366
and because

632
00:45:43,366 --> 00:45:45,006
they have different fundamental properties

633
00:45:45,006 --> 00:45:47,026
the idea is that

634
00:45:47,026 --> 00:45:49,286
one way is by going to a hash based algorithm

635
00:45:49,286 --> 00:45:50,886
you'd be able to stay quantum proof

636
00:45:50,886 --> 00:45:58,746
this is also why in bitcoin today if you deposit bitcoin to an address and it's not taproot and

637
00:45:58,746 --> 00:46:01,946
it's not the original paid a public key any of the other ones that have hashed addresses

638
00:46:01,946 --> 00:46:07,986
the actual data that sits on the blockchain is a hash of a public key or a hash of a script and

639
00:46:07,986 --> 00:46:12,206
because it's just a hash it's not something that a quantum computer would be able to reverse

640
00:46:12,206 --> 00:46:16,446
engineer and this goes into one last point about this which between a long range and a short range

641
00:46:16,446 --> 00:46:21,086
quantum attack. A long range quantum attack is I have years or I have lots of heads up time to be

642
00:46:21,086 --> 00:46:27,246
able to look at a public key and grind away and find the private key. A short range attack is

643
00:46:27,246 --> 00:46:31,946
I have a very small window to be able to reverse engineer the public private key. And the way this

644
00:46:31,946 --> 00:46:36,726
intersects with Bitcoin is that if you have an address that you haven't spent from before,

645
00:46:36,806 --> 00:46:40,926
you've never exposed the public key to the blockchain, broadcast and send, there's roughly

646
00:46:40,926 --> 00:46:46,126
like a 10-ish minute window where someone could in theory take a quantum computer, attack your

647
00:46:46,126 --> 00:46:49,446
public key, find your private key and try and double spend you.

648
00:46:50,066 --> 00:46:53,886
That's a way harder problem because it takes something that's in the domain of

649
00:46:53,886 --> 00:46:56,026
five years to try and guess it because it's sitting there.

650
00:46:59,146 --> 00:47:01,926
And while I think that it's an important

651
00:47:01,926 --> 00:47:05,786
thing that Bitcoin development should be able to do, a quantum computer hasn't been able to factor

652
00:47:05,786 --> 00:47:09,146
the number 21 on its own yet without a lot of accelerated help

653
00:47:09,146 --> 00:47:13,186
embedded learnings in the programs themselves that are kind of baking in the answer.

654
00:47:13,186 --> 00:47:22,986
So I kind of duly caution that I think this is important existential for Bitcoin to be resolved in a timely manner over the next five years to have a plan.

655
00:47:23,506 --> 00:47:26,566
But it's not something that I think tomorrow is going to bring Bitcoin.

656
00:47:27,566 --> 00:47:29,306
And I totally agree.

657
00:47:29,606 --> 00:47:30,966
I totally agree with that.

658
00:47:30,966 --> 00:47:42,846
I am personally going to – I'm going to unfortunately for myself, but I am going to ramp up my ability to understand and explain this.

659
00:47:43,186 --> 00:47:53,286
I, I, it's very hard to explain why you're not worried about it without really, really being, um, strong in the details.

660
00:47:53,466 --> 00:48:13,086
So if you, it's the, the price of being able to explain it is the time spent understanding it, even though you kind of think it's a waste of time because it's not a real threat, but it's just more of, um, I have created a textbook where, um, and this came at, I feel like I had the idea for this after our first episode, but I finally just did it.

661
00:48:13,186 --> 00:48:20,306
And it basically said, take me from the basic arithmetic to understanding Shor's algorithm and Grover's algorithm.

662
00:48:21,766 --> 00:48:27,426
And what it ended up doing was it didn't just create, you know, it didn't just take me through the math.

663
00:48:27,526 --> 00:48:34,466
It actually sort of understood the various, everything you just said and what that means for Bitcoin.

664
00:48:34,686 --> 00:48:37,186
And so it's not a yes or no.

665
00:48:37,226 --> 00:48:38,086
It's quantum dangerous.

666
00:48:38,086 --> 00:48:44,826
It's like there's a stack of low-hanging fruit that goes up to high-hanging fruit.

667
00:48:45,486 --> 00:48:49,006
And it's probably worthwhile to understand that stack, right?

668
00:48:49,366 --> 00:49:02,546
It's probably worthwhile to understand what makes Satoshi's coins vulnerable in a way that what makes, you know, if you're behind Taproot and you've never exposed your public key.

669
00:49:02,906 --> 00:49:04,706
Like it's good to understand these things.

670
00:49:04,706 --> 00:49:13,206
and um so i plan to go on this journey with this textbook i created and um be able to hopefully

671
00:49:13,206 --> 00:49:22,046
explain at least you know some in some fashion try to get us through the way i was um the way i

672
00:49:22,046 --> 00:49:27,626
explained this to somebody they were asking me like i was asked to be on a panel about quantum

673
00:49:27,626 --> 00:49:32,646
and then they were like can you actually do that and i said to be honest i'm a three out of ten on

674
00:49:32,646 --> 00:49:37,926
this. Okay. But the people who are talking about quantum are like a five out of 10 and the ones

675
00:49:37,926 --> 00:49:44,966
think they're tens. And I can take the ones, I can tell, I can show the ones what a five is.

676
00:49:46,186 --> 00:49:50,826
And then the guy was like, you can't explain anything, can you? You're really bad at this.

677
00:49:52,646 --> 00:49:56,806
But I mean, I thought that was a pretty solid explanation for what the, for, you know, what

678
00:49:56,806 --> 00:50:04,286
the space is here right no for sure and so but all this i i'm it's fine talking about this here

679
00:50:04,286 --> 00:50:09,606
because it you know at some point we're doing you know we're hurting our brains to try to understand

680
00:50:09,606 --> 00:50:14,226
something it's good to be able to say well why the hell am i doing this it's good to you know

681
00:50:14,226 --> 00:50:21,926
as much as i will generally shit on that notion i mean i think in general we should be willing to

682
00:50:21,926 --> 00:50:26,226
learn without always having to be like well why am i learning this why am i learning this is the

683
00:50:26,226 --> 00:50:34,246
reason why math is censored it's censored in our own minds in our own sense of time but to the

684
00:50:34,246 --> 00:50:40,106
extent you can motivate it we should and that's why i think it's it's okay when we talk about

685
00:50:40,106 --> 00:50:43,986
discrete logarithm problem and you know there's people that are just going to be like well why

686
00:50:43,986 --> 00:50:48,786
would i bother learning this if it's going to be essentially by the time i'm done learning it by

687
00:50:48,786 --> 00:50:53,446
the time my brain understands it it's going to be destroyed by a quantum computer right and i would

688
00:50:53,446 --> 00:50:57,546
just say and then oh so what i'm gonna and then once i get to that point i'm gonna have to now

689
00:50:57,546 --> 00:51:03,146
learn what a quantum computer does and doesn't do um and i'd say you know yeah it's all this is if

690
00:51:03,146 --> 00:51:12,086
if you're all in like uh rob and i are yeah you better kind of better do it you know if you're a

691
00:51:12,086 --> 00:51:17,746
tourist you're fully in it if you're fully in it man there's really no other way i threw it

692
00:51:17,746 --> 00:51:23,286
i mean the one thing i'll say about coblets he's all in and his thing he's all in with whatever

693
00:51:23,286 --> 00:51:28,166
he's doing and he's going to pursue that rob and i are all in here on bitcoin and we're gonna

694
00:51:28,166 --> 00:51:35,506
we're gonna tell you guys we're gonna hopefully pave the path for those who really feel like they

695
00:51:35,506 --> 00:51:42,826
have to understand this at a basic level and um you know do you have to no but you probably we

696
00:51:42,826 --> 00:51:47,486
probably should and i think it's again gonna continue we're gonna continue to do this so now

697
00:51:47,486 --> 00:51:54,906
my, my, what's the second meaning? I love the discrete log name because discrete, we talked

698
00:51:54,906 --> 00:52:00,646
about discrete meaning like fixed in time, not continuous. And then I love that discrete also

699
00:52:00,646 --> 00:52:10,166
means secret, you know, like that to me, the, that's, that's, that's of the God world right

700
00:52:10,166 --> 00:52:14,706
there of giving us this thing that means both of those things, you know, and it's the discreteness

701
00:52:14,706 --> 00:52:15,666
that makes it a secret.

702
00:52:17,046 --> 00:52:17,526
Right.

703
00:52:18,226 --> 00:52:18,686
Right.

704
00:52:19,446 --> 00:52:19,806
The,

705
00:52:19,946 --> 00:52:21,526
I'm looking,

706
00:52:22,026 --> 00:52:23,606
I'm looking through this.

707
00:52:23,746 --> 00:52:26,226
We're at a dangerous time of the episode.

708
00:52:26,606 --> 00:52:27,006
Yeah.

709
00:52:27,186 --> 00:52:29,386
I don't want to overshoot into a clearing

710
00:52:29,386 --> 00:52:31,726
and then leave everyone to go figure it out.

711
00:52:32,746 --> 00:52:33,266
For,

712
00:52:33,866 --> 00:52:36,366
I'm flipping through the study guide right now.

713
00:52:36,846 --> 00:52:38,426
I might cut five minutes out of

714
00:52:38,426 --> 00:52:39,466
where I stepped in shit.

715
00:52:40,406 --> 00:52:41,126
What was that?

716
00:52:41,566 --> 00:52:43,306
I might cut the five minutes out

717
00:52:43,306 --> 00:52:44,146
where I stepped in shit.

718
00:52:44,146 --> 00:52:46,966
It wasn't that bad.

719
00:52:48,966 --> 00:52:54,246
Looking through the study guide here, the curve over the reels, the signature.

720
00:52:56,386 --> 00:52:59,406
I know you did a little bit of work, too, on Bitcoin transactions in general.

721
00:52:59,586 --> 00:53:05,626
Is there something you want to take a little detour on as it relates to Bitcoin transactions themselves?

722
00:53:06,746 --> 00:53:12,266
We could always go back to our buddy, the nonces, the generating of a signature.

723
00:53:12,266 --> 00:53:16,946
We've gone over and creating a private key.

724
00:53:16,946 --> 00:53:19,346
We've done the public key multiple previously.

725
00:53:19,346 --> 00:53:21,346
We can do that again for emphasis.

726
00:53:21,346 --> 00:53:23,646
I think let's save those.

727
00:53:23,646 --> 00:53:29,026
So I feel like when – what we've been doing in arithmetic, we mention these things that

728
00:53:29,026 --> 00:53:33,626
are at the end because – to motivate us to say this is the line we're – this is

729
00:53:33,626 --> 00:53:36,666
the point we're walking to on this line.

730
00:53:36,666 --> 00:53:40,826
I don't know if I want to get too deep into the protocol stuff.

731
00:53:40,826 --> 00:53:45,706
I think we call it out so that we say this is why we're learning what a logarithm is.

732
00:53:45,926 --> 00:53:46,226
Right.

733
00:53:46,646 --> 00:53:46,886
Right.

734
00:53:47,486 --> 00:53:54,446
So maybe if we're getting gun shy about sort of what we think is the last arithmetic.

735
00:53:56,326 --> 00:53:56,946
I don't know.

736
00:53:57,026 --> 00:53:57,766
I think we could do it.

737
00:53:57,786 --> 00:53:58,366
I think we could do it.

738
00:53:58,486 --> 00:53:59,186
I think we should do it.

739
00:53:59,906 --> 00:54:00,266
Okay.

740
00:54:00,946 --> 00:54:07,886
So there's the last – I feel like the last piece of arithmetic that we need here is –

741
00:54:07,886 --> 00:54:13,766
unfortunately it's a little it's a little you know it's a little it's a little hairy but it's

742
00:54:13,766 --> 00:54:19,046
not that hairy but it's something so in mod you know we talked let's go back to modular

743
00:54:19,046 --> 00:54:28,566
modular arithmetic okay something i don't think we really discussed that deeply is that um the

744
00:54:28,566 --> 00:54:35,386
properties of modular arithmetic meaning so like if i have nine mod seven then that equals two

745
00:54:35,386 --> 00:54:45,486
right and if i have um 10 mod 7 and that equals 3 that 9 plus 10 is always going to equal

746
00:54:45,486 --> 00:54:50,466
2 plus 3 i hate that i had to say this so i'm going to try to really draw this line

747
00:54:50,466 --> 00:54:58,866
9 plus 10 9 being 2 right in mod 7 10 being 3 mod 7 that sum is always going to be

748
00:54:58,866 --> 00:55:03,426
5 so in other words 9 plus 10 is always going to be interchangeable with 2 plus 3

749
00:55:03,426 --> 00:55:08,306
always, always, always, always. Right. And I can say, I know 19 mod seven,

750
00:55:08,726 --> 00:55:10,806
God, please God tell me it's five. Right.

751
00:55:12,946 --> 00:55:15,566
It is so like under addition,

752
00:55:15,966 --> 00:55:20,426
it's like what we're trying to say here is this is back to like the abstractness

753
00:55:20,426 --> 00:55:24,006
and the groups and the structure, but like the operation is always preserved.

754
00:55:24,546 --> 00:55:29,326
No matter what, no matter which version I'm doing. Right.

755
00:55:29,326 --> 00:55:33,146
If I'm doing the regular arithmetic or if I'm doing, you know,

756
00:55:33,146 --> 00:55:36,846
But no matter what order I'm doing it in, if I take the mod first or if I wait till the

757
00:55:36,846 --> 00:55:42,166
end to take the modulus, it's always going to get to the same answer.

758
00:55:42,166 --> 00:55:43,166
And that's true of addition.

759
00:55:43,166 --> 00:55:45,006
It's true of subtraction.

760
00:55:45,006 --> 00:55:47,706
It's true of multiplication.

761
00:55:47,706 --> 00:55:53,466
So this is something that I've spent probably months in abstract algebra textbooks and number

762
00:55:53,466 --> 00:55:54,926
theory textbooks proving.

763
00:55:54,926 --> 00:55:58,766
But it's all very – it's just good to – you want to know that these things are

764
00:55:58,766 --> 00:56:02,886
all true in modular arithmetic.

765
00:56:02,886 --> 00:56:08,646
might seem like an unnecessary thing to say, but you can always count on the properties.

766
00:56:09,146 --> 00:56:17,106
This operation, this idea of using a modulus is going to work. It's going to work no matter

767
00:56:17,106 --> 00:56:24,166
what we're doing. Okay. And so things get a little more advanced when we're talking about

768
00:56:24,166 --> 00:56:26,286
squares.

769
00:56:27,266 --> 00:56:29,266
So the first question is

770
00:56:29,266 --> 00:56:31,566
why do we care

771
00:56:31,566 --> 00:56:33,946
if something is a square or not?

772
00:56:33,986 --> 00:56:35,286
And I should say this statement,

773
00:56:35,526 --> 00:56:38,246
we now also care

774
00:56:38,246 --> 00:56:39,706
that a number is a square. Why?

775
00:56:40,606 --> 00:56:41,646
And that is because

776
00:56:41,646 --> 00:56:43,966
of what the Bitcoin elliptic curve

777
00:56:43,966 --> 00:56:45,006
equation is.

778
00:56:45,426 --> 00:56:47,206
It says y squared equals something.

779
00:56:48,566 --> 00:56:49,086
Right?

780
00:56:49,766 --> 00:56:52,166
It equals x cubed plus 7

781
00:56:52,166 --> 00:56:53,626
but it really is just a matter of this.

782
00:56:53,626 --> 00:57:03,086
y squared equals something. So that thing, that point on your curve is a square. Okay.

783
00:57:03,346 --> 00:57:10,026
That's just really, by definition, it is a square. Okay. Now, why does that matter? Okay. Now it's

784
00:57:10,026 --> 00:57:19,226
not a, it's not a geometric square. It's a number that can be, that can be gotten by taking a number

785
00:57:19,226 --> 00:57:26,286
multiplied by itself in the in the modulus in the modular space modulo um in bitcoin's case

786
00:57:26,286 --> 00:57:37,306
modulo two to the 256 minus two to the 32 minus 977 right so the point the answer is is a square

787
00:57:37,306 --> 00:57:41,906
that that so that's a little this is why we were a little gun shy of jumping right into this because

788
00:57:41,906 --> 00:57:46,946
it's like that you know i don't feel great about how this is coming we're about to do an ensemble

789
00:57:46,946 --> 00:57:52,186
of combining a bunch of concepts we've gone up to up to this point from modular arithmetic world

790
00:57:52,186 --> 00:57:58,666
from groups from the equation of y squared equals x cubed plus seven which is the elliptic curve

791
00:57:58,666 --> 00:58:04,566
function we're about to smash all of those together and i think the first thing to start

792
00:58:04,566 --> 00:58:10,286
with with your point about y squares are important is that very specifically by definition of the way

793
00:58:10,286 --> 00:58:15,946
the elliptic curve algorithm is is y squared equals x cubed plus seven so that means that

794
00:58:15,946 --> 00:58:18,766
it will must necessarily always be a square

795
00:58:18,766 --> 00:58:20,946
for it to satisfy the elliptic curve function.

796
00:58:21,706 --> 00:58:25,206
Now, this is where we're dancing around

797
00:58:25,206 --> 00:58:27,406
the last leap here of a concept.

798
00:58:30,226 --> 00:58:31,726
Is it important, by the way,

799
00:58:31,826 --> 00:58:34,786
is it helpful for us to just know it's a square

800
00:58:34,786 --> 00:58:38,206
versus having to know how to...

801
00:58:38,206 --> 00:58:40,006
I don't think we're ever really asked

802
00:58:40,006 --> 00:58:41,086
to take the square root of anything.

803
00:58:41,206 --> 00:58:42,646
We just have to know how to know

804
00:58:42,646 --> 00:58:44,586
a number is a perfect square and on our curve.

805
00:58:45,946 --> 00:58:47,466
That distinction.

806
00:58:48,326 --> 00:58:58,826
So that's part of why it's helpful and it's part of why – that's part of the power of elliptic curve ECDSA.

807
00:58:59,386 --> 00:58:59,906
Yes.

808
00:59:00,526 --> 00:59:03,666
Right? Or sorry, ECDLP I should say.

809
00:59:03,966 --> 00:59:04,326
Yes.

810
00:59:05,866 --> 00:59:07,386
DLP meaning discrete log problem.

811
00:59:07,606 --> 00:59:10,726
So what we're solving is the elliptic curve discrete log problem for Bitcoin.

812
00:59:10,726 --> 00:59:15,146
Right. And what we need to know, the ultimate point of this is if the point is on the curve or not.

813
00:59:15,146 --> 00:59:19,846
If the number is a square on the curve, if not, then it's not on the curve.

814
00:59:20,306 --> 00:59:26,786
So like what I want to just illustrate real quick is definitely got to make a video for this.

815
00:59:27,006 --> 00:59:34,766
What I want to illustrate is just how, okay, on our number system that we love using, like the reels or the, you know, really we'll call it the integers, right?

816
00:59:35,486 --> 00:59:40,946
We have a pretty good idea for, we know that not all numbers are perfect squares, right?

817
00:59:40,966 --> 00:59:42,246
I'm going to say a bunch of obvious things.

818
00:59:42,246 --> 00:59:50,426
We know what they are, 1, 4, 9, 16, 25, 36, 49, et cetera, right?

819
00:59:50,446 --> 00:59:53,866
We know that not all numbers are perfect squares.

820
00:59:54,126 --> 00:59:57,966
Only a certain subset of our numbers are perfect squares, right?

821
00:59:59,246 --> 01:00:05,226
And so now I'm going to go back and bring to bear something we talked about in the past,

822
01:00:05,246 --> 01:00:11,086
which is the fact that a finite field modulo P is a different number system, okay?

823
01:00:11,086 --> 01:00:17,306
remember that two numbers can multiply to each other to equal one we don't do division

824
01:00:17,306 --> 01:00:23,886
to to get one or to do inverses we do multiplications to get inverses right so

825
01:00:23,886 --> 01:00:32,046
and remember like if if um if the mod is five right two is the inverse of three because two

826
01:00:32,046 --> 01:00:40,146
times three is equals six i e one mod five right so like in the so now we're in this number system

827
01:00:40,146 --> 01:00:44,666
that's akin to the integers, but it's called the finite field modulo p.

828
01:00:45,726 --> 01:00:46,266
Okay, right?

829
01:00:47,546 --> 01:00:49,806
And in this number system,

830
01:00:51,306 --> 01:00:57,746
it's finding a square is, you would say, is probably similar, right?

831
01:00:57,866 --> 01:01:01,346
Not all your numbers are perfect squares.

832
01:01:02,006 --> 01:01:05,226
And just so by way of an example, mod 7, right?

833
01:01:06,326 --> 01:01:10,126
This I did do this morning, so I'm ready to hit you with it.

834
01:01:10,146 --> 01:01:14,366
it, but you may want to like, you may want to pause at this point, get a pen and paper and do

835
01:01:14,366 --> 01:01:21,866
this with me. It's not hard. One squared equals one, two squared equals four. So those so far

836
01:01:21,866 --> 01:01:26,026
are the easy ones because mod seven, you don't have to, we haven't exceeded the modulus yet,

837
01:01:26,246 --> 01:01:32,746
right? Three squared equals nine, which mod seven is two. Okay. So now I have, I have one,

838
01:01:32,746 --> 01:01:50,759
4 and 2 is perfect squares in mod 7 space So now I going to move on I have three more to do Four squared which is 16 That two also So now I just okay so two is a perfect that no new information I already knew

839
01:01:50,839 --> 01:01:56,899
I already knew two was a perfect square, but you know what? I actually, you know, in the, in the

840
01:01:56,899 --> 01:02:04,859
reels, perfect square has two solutions, right? Well, this finite field also has two solutions,

841
01:02:04,859 --> 01:02:11,259
right so to the square root of two in the finite field modulo seven it's going to be

842
01:02:11,259 --> 01:02:19,119
three and four both of those numbers when squared give you the answer to okay um moving on to five

843
01:02:19,119 --> 01:02:28,059
five squared equals four so which was two squared also so now two and five are the numbers in the

844
01:02:28,059 --> 01:02:34,139
finite field that give you four when you square them and finally six squared equals one and so

845
01:02:34,139 --> 01:02:40,339
one and six are now the two answers say what what two numbers so instead we're not doing plus or

846
01:02:40,339 --> 01:02:46,579
minus anymore like we did in the real space right we're doing the two numbers that multiply to each

847
01:02:46,579 --> 01:02:53,499
you know there are two distinct numbers that do this there's still two answers right which is cool

848
01:02:53,499 --> 01:02:59,679
right it's like oh algebra still says the same stuff right there's still two answers to a square

849
01:02:59,679 --> 01:03:08,539
root. So just again, one, two, and four are now the perfect squares in this system, which

850
01:03:08,539 --> 01:03:15,059
sorry as an aside, if this can confuse anybody, but it's too cool not to mention that they

851
01:03:15,059 --> 01:03:21,459
add up to seven. You won't see this again. You won't see it with another modulus. It's

852
01:03:21,459 --> 01:03:25,259
just, okay, so one, two, but it's a good way to remember one, two, and four are your perfect

853
01:03:25,259 --> 01:03:27,599
squares in modulo 7.

854
01:03:30,039 --> 01:03:31,279
So if the

855
01:03:31,279 --> 01:03:33,299
Bitcoin elliptic curve

856
01:03:33,299 --> 01:03:34,979
was modulo 7, right,

857
01:03:35,219 --> 01:03:37,359
you would want to make sure

858
01:03:37,359 --> 01:03:39,099
that 1, 2, and 4

859
01:03:39,099 --> 01:03:41,159
were the numbers on your, were the points on your

860
01:03:41,159 --> 01:03:41,979
curve and not

861
01:03:41,979 --> 01:03:44,379
2, 5, and 6.

862
01:03:45,919 --> 01:03:47,219
Right? Because you

863
01:03:47,219 --> 01:03:49,079
got fake, you got something fake.

864
01:03:49,899 --> 01:03:50,859
Yeah, not on the curve.

865
01:03:51,259 --> 01:03:53,079
And it's roughly half when you go

866
01:03:53,079 --> 01:03:54,619
across the full actual Bitcoin.

867
01:03:55,259 --> 01:03:59,579
curve, roughly half the points along the curve have perfect squares, no other half.

868
01:04:00,619 --> 01:04:05,179
Well, actually, it's not that I'm going to correct you. I'm going to enhance your answer.

869
01:04:05,419 --> 01:04:11,579
It's whatever your prime is. So if you remember, a finite field modulo P has P minus, well,

870
01:04:11,579 --> 01:04:20,519
has P minus one elements, right? So modulo seven has six elements. So it is half of six.

871
01:04:20,519 --> 01:04:23,639
So it's P minus 1 over – it is half of P minus 1.

872
01:04:23,839 --> 01:04:25,699
It's exactly – it's always the answer.

873
01:04:27,399 --> 01:04:33,599
And P in this case is the 2 to the 256 minus 2 to the 232 minus 2 to the 16 minus 977.

874
01:04:34,179 --> 01:04:36,419
So basically a massive number.

875
01:04:36,859 --> 01:04:40,079
And by the way, it has to be half because of what I just said.

876
01:04:40,519 --> 01:04:44,319
Because two numbers always map into one square.

877
01:04:44,759 --> 01:04:45,479
Right, right.

878
01:04:45,839 --> 01:04:47,059
So it's got to be half.

879
01:04:47,059 --> 01:04:49,939
And you can just even look at the curve and just think about that too.

880
01:04:49,939 --> 01:04:56,279
that every x point along the curve has a lower half and an upper half above and below the number one.

881
01:04:56,579 --> 01:04:57,519
We call that subjective.

882
01:04:57,999 --> 01:05:00,699
We call it, you know, the one also called subjective in algebra,

883
01:05:00,919 --> 01:05:03,759
which some people listening may appreciate that I said that.

884
01:05:04,159 --> 01:05:04,759
Most people didn't.

885
01:05:05,279 --> 01:05:08,599
And so if you think about this too, right, a point on a curve,

886
01:05:08,699 --> 01:05:10,199
you have an x and a y coordinate, right?

887
01:05:10,459 --> 01:05:12,499
Just as one thing, just to kind of call out.

888
01:05:13,079 --> 01:05:14,839
Rob is drawing that with his hands, by the way.

889
01:05:15,019 --> 01:05:15,519
Yes, I am.

890
01:05:15,599 --> 01:05:16,639
I'm drawing the axi, right?

891
01:05:16,639 --> 01:05:18,759
So you have a grid of, you know,

892
01:05:18,759 --> 01:05:22,439
just think about if you've ever been in like a math class with like an xy axis in the grid

893
01:05:22,439 --> 01:05:32,119
this curve along any given x point will be two y points which actually you're actually able to

894
01:05:32,119 --> 01:05:38,559
infer basically a you're able to compress the actual public key at any point if i just give you

895
01:05:38,559 --> 01:05:46,159
x and then i give you one extra byte for whether it's the positive or the negative side of the top

896
01:05:46,159 --> 01:05:50,239
or the bottom part of the graph of the electric curve.

897
01:05:51,319 --> 01:05:52,819
With the exception of Taproot,

898
01:05:54,899 --> 01:05:58,099
Taproot, this is called remove the parity bit.

899
01:05:58,519 --> 01:06:01,859
You're able to, in pre-Taproot, it's a 33-byte key,

900
01:06:01,959 --> 01:06:03,419
32 bytes for your X-coordinate,

901
01:06:03,979 --> 01:06:05,799
and then one byte for positive or negative.

902
01:06:07,079 --> 01:06:12,319
The way it works in Taproot is you just have the 32 bytes,

903
01:06:12,319 --> 01:06:14,079
and then there's an assumption that's made.

904
01:06:14,079 --> 01:06:18,359
and it always chooses an even value for one.

905
01:06:19,719 --> 01:06:20,239
Interesting.

906
01:06:20,799 --> 01:06:21,319
Yes.

907
01:06:21,859 --> 01:06:23,459
Taproot really screwed some things up.

908
01:06:23,559 --> 01:06:24,779
That's why everybody hates it, right?

909
01:06:25,279 --> 01:06:30,559
Well, it's funny because as an aside for Bitcoin protocol development,

910
01:06:30,859 --> 01:06:33,899
this was like a version of byte pinching where we said,

911
01:06:33,899 --> 01:06:38,659
oh, we can save one byte across all Taproot transactions for better scaling

912
01:06:38,659 --> 01:06:42,459
because we can just infer that.

913
01:06:42,459 --> 01:06:47,739
And it turns out for designing more advanced covenant-like applications,

914
01:06:47,899 --> 01:06:52,779
sometimes you want the other point of the key to do stuff for like unordered exits.

915
01:06:53,139 --> 01:06:54,819
And we should have reared it on at some point.

916
01:06:54,879 --> 01:06:55,799
You can go deeper into this.

917
01:06:56,379 --> 01:06:59,559
One of the things where we wanted to save a byte because that would help with scaling.

918
01:06:59,779 --> 01:07:00,399
It's less data.

919
01:07:00,819 --> 01:07:03,199
But it ended up having way worse trade-offs.

920
01:07:05,199 --> 01:07:08,939
I went into a very immature space when you said byte-pinching.

921
01:07:09,059 --> 01:07:09,399
I'm sorry.

922
01:07:12,459 --> 01:07:13,619
I thought about Marv Albert.

923
01:07:17,939 --> 01:07:22,559
Anyhow, you know, this is like my defense mechanism

924
01:07:22,559 --> 01:07:26,099
because this is how I feel like to anesthetize you guys a little bit

925
01:07:26,099 --> 01:07:28,759
this very difficult conversation.

926
01:07:29,859 --> 01:07:35,819
But the ability to know if something is a square is actually,

927
01:07:36,059 --> 01:07:41,499
if you do enough math, if you tackle it hard enough, okay,

928
01:07:41,499 --> 01:08:02,339
And it's very tackleable. But if you do it, there's a very then there becomes Euler basically, like as Euler and Gauss gave very elegant ways to, you know, for computers that didn't exist yet to determine whether or not something is a is a perfect square or not.

929
01:08:02,339 --> 01:08:10,479
And, you know, Koblis' book is pretty good, but I think you want to go to elementary number theory textbooks.

930
01:08:11,379 --> 01:08:17,319
This is like the final boss of the easy parts of a number theory textbook.

931
01:08:17,819 --> 01:08:21,139
And it's also the final boss of the easy parts of our explanation.

932
01:08:22,999 --> 01:08:26,639
But if you really want to grok this, you're going to have to do some problems.

933
01:08:26,779 --> 01:08:27,519
You're going to have to sit there.

934
01:08:27,519 --> 01:08:33,799
And the nice thing is you can sit and do what I just did, mod 7, do it mod 11.

935
01:08:34,619 --> 01:08:38,859
It ends up that, like, I just found out this fun fact today,

936
01:08:39,259 --> 01:08:42,719
but it ends up in determining a quadratic residue.

937
01:08:43,319 --> 01:08:48,699
The algorithm needs to know if your prime is either 1 mod 4 or 3 mod 4.

938
01:08:48,939 --> 01:08:50,579
So what does that mean?

939
01:08:51,539 --> 01:08:53,019
5 is 1 mod 4.

940
01:08:53,199 --> 01:08:56,099
Every prime number is either 1 mod 4 or 3 mod 4.

941
01:08:56,099 --> 01:08:58,319
Every odd number is 1 mod 4, right?

942
01:08:58,399 --> 01:09:02,239
Every odd number when divided by 4 is going to be either 1 or 3, okay?

943
01:09:02,319 --> 01:09:02,499
Right?

944
01:09:02,579 --> 01:09:05,079
So just slow that down.

945
01:09:05,519 --> 01:09:11,919
So, you know, 5, 13, 17, those are 1 mod 4.

946
01:09:13,779 --> 01:09:19,179
3, obviously, 7, 11, these are 3 mod 4s.

947
01:09:19,199 --> 01:09:20,859
And they behave differently in this algorithm.

948
01:09:20,859 --> 01:09:28,199
And it just, it turns out that the numbers used in Bitcoin are all three mod four.

949
01:09:28,319 --> 01:09:29,119
I found this out today.

950
01:09:29,199 --> 01:09:31,219
That is, and that blew my mind.

951
01:09:31,459 --> 01:09:34,919
And I think it's another property that makes, yeah,

952
01:09:35,019 --> 01:09:40,299
just probably either makes the computation work faster in the way we want.

953
01:09:40,759 --> 01:09:42,099
That you're on mute.

954
01:09:42,239 --> 01:09:44,039
God, how long, how long have you been talking, dude?

955
01:09:44,379 --> 01:09:45,799
No, no, I just, I literally just started.

956
01:09:45,799 --> 01:09:52,659
I mean, dude, God exists to give me a podcast partners that just talk on mute so that I can just talk forever.

957
01:09:52,939 --> 01:09:53,619
That's right.

958
01:09:54,019 --> 01:09:54,339
That's right.

959
01:09:54,539 --> 01:09:55,199
No, we need you, man.

960
01:09:55,839 --> 01:09:56,179
No, no.

961
01:09:56,259 --> 01:09:56,899
This is what I was going to say.

962
01:09:57,139 --> 01:10:12,679
Like to this point specifically, like it's – the shortcut that's allowed here is that you're able to do a square root in just one single exponentiation, right?

963
01:10:12,679 --> 01:10:15,659
You're just able to quickly – and you can think about it that way.

964
01:10:15,799 --> 01:10:18,359
like the square root of something is taking it to the one half power.

965
01:10:18,839 --> 01:10:24,499
This makes it really short and straightforward to be able to do this like

966
01:10:24,499 --> 01:10:25,519
arithmetic. Right.

967
01:10:28,039 --> 01:10:32,419
The, as a, as a recap, like, like, cause you,

968
01:10:32,479 --> 01:10:34,819
you said the term, but we didn't talk about it more,

969
01:10:34,879 --> 01:10:36,179
but like a quadratic residue,

970
01:10:36,179 --> 01:10:40,899
which is kind of like the last thing that we're putting forward in this

971
01:10:40,899 --> 01:10:45,379
episode to kind of close out this section is that, you know,

972
01:10:45,799 --> 01:10:50,019
This inversion we're talking about that I mentioned earlier,

973
01:10:50,519 --> 01:10:51,819
like if it's on the point or not,

974
01:10:51,899 --> 01:10:53,239
requires it being a quadratic radius.

975
01:10:53,439 --> 01:10:54,839
It requires it being a perfect square.

976
01:10:55,579 --> 01:11:01,019
And being able to reduce that into a quick one-step calculation

977
01:11:01,019 --> 01:11:05,239
is really important for being able to...

978
01:11:05,239 --> 01:11:07,179
You just square and multiply the number,

979
01:11:07,499 --> 01:11:10,279
and we can do that pretty quickly from the math that we've already done.

980
01:11:11,899 --> 01:11:14,879
I'm now starting to crash and burn a little bit here from where I was going to go.

981
01:11:14,879 --> 01:11:19,339
So in number theory, there's something called the Legendre symbol.

982
01:11:20,299 --> 01:11:23,179
And it's just you got to give it to Gauss again.

983
01:11:23,299 --> 01:11:29,139
This guy is just all over everything we're able to do here.

984
01:11:29,139 --> 01:11:44,499
But he's able to abstract all of essentially what we just talked about and all of that pain and trying to figure out what number multiplies itself to get us another number into a symbol with a bunch of rules.

985
01:11:44,879 --> 01:11:52,199
that and an algorithm so i'm not sure where you were going there but like this notion of a quad we

986
01:11:52,199 --> 01:11:57,259
called it a sorry we called it a quadratic residue and i don't want jargon to get confusing quadratic

987
01:11:57,259 --> 01:12:02,799
is always like a square quadratic equation is something that x squared plus ax squared plus bx

988
01:12:02,799 --> 01:12:09,719
plus c that's quadratic equation so when we talk about quadratics we're talking about square roots

989
01:12:09,719 --> 01:12:14,719
The word residue is always from modular arithmetic, the remainder.

990
01:12:15,799 --> 01:12:26,279
So when we talk about quadratic residues, we're talking about square roots specifically in modular arithmetic that turn out to always be whole numbers, which is awesome.

991
01:12:28,739 --> 01:12:31,719
And that is, it's not like super necessary.

992
01:12:32,759 --> 01:12:34,179
It's not super necessary.

993
01:12:34,179 --> 01:12:42,659
But I think in order to have a rich conversation here, it's important not to leave out that the point on the elliptic curve is a square by definition.

994
01:12:43,219 --> 01:12:50,259
And we do need to – there's richness here in studying it.

995
01:12:50,499 --> 01:12:53,859
So if you've gotten this far, good on you.

996
01:12:55,439 --> 01:12:56,039
Yeah.

997
01:12:56,039 --> 01:13:06,779
um i'm thinking through the on where to take it from here but this whole quadratic residue concept

998
01:13:06,779 --> 01:13:12,979
um quick heuristic of making sure your point's on the curve like i mentioned earlier when you're

999
01:13:12,979 --> 01:13:17,999
putting your compressed public key being able to quickly just get the exact point because you need

1000
01:13:17,999 --> 01:13:24,339
the exact point if you're going to verify the public key against a signature so this is just

1001
01:13:24,339 --> 01:13:29,939
another like mathematical shortcut and computational shortcut that allows the whole engine to run

1002
01:13:29,939 --> 01:13:36,279
smoothly. Yeah. And I think, I think what we're trying to do is create a little list in our heads

1003
01:13:36,279 --> 01:13:42,439
of which I've mentioned this before, but I love this construct. What should we have asked the

1004
01:13:42,439 --> 01:13:49,079
manager about when we were at the Bitcoin store buying our Bitcoin? You know, we definitely should

1005
01:13:49,079 --> 01:13:54,979
have asked about inverses and we definitely should have asked is about is it a square does it exist

1006
01:13:54,979 --> 01:14:00,039
on my curve yeah oh this is this is really important too because everyone is using this

1007
01:14:00,039 --> 01:14:04,119
if you've ever spent to receive bitcoin you're running software that's doing this stuff that

1008
01:14:04,119 --> 01:14:09,839
we're talking about and if you've ever if you have bitcoin on the network and you just see an address

1009
01:14:09,839 --> 01:14:11,959
how do you know that address belongs to you

1010
01:14:11,959 --> 01:14:20,459
right what you can do is you could take your secret you could take your secret key you could

1011
01:14:20,459 --> 01:14:24,119
map that to a public key you could then look at that public key and then you could look and say

1012
01:14:24,119 --> 01:14:28,859
okay is this tapper is this seglet whatever and you could say okay well here's my key let me

1013
01:14:28,859 --> 01:14:34,239
compress it and then let me hash it and then do the base 58 encoding and that matchy matches my

1014
01:14:34,239 --> 01:14:38,739
address right and if you've held bitcoin for a long time you've never thought through like those

1015
01:14:38,739 --> 01:14:44,619
steps that's something that like you said when when you say go to the bitcoin store

1016
01:14:44,619 --> 01:14:50,859
anytime you go to the bitcoin blockchain and you receive money or send money you're going to the

1017
01:14:50,859 --> 01:14:55,239
bitcoin store you're doing this stuff so like you want to find out before you deposit money

1018
01:14:55,239 --> 01:15:00,219
that my point is actually on the curve you don't want to find out after the fact because there's

1019
01:15:00,219 --> 01:15:05,219
no do-overs no reviews right well that's the whole thing and so this is i've been locked in

1020
01:15:05,219 --> 01:15:06,579
on this very subject.

1021
01:15:06,699 --> 01:15:07,759
This may be what we close on,

1022
01:15:07,879 --> 01:15:11,219
but I've been locked in on...

1023
01:15:12,959 --> 01:15:14,559
Maybe it's a little provocative

1024
01:15:14,559 --> 01:15:18,119
and maybe I'm being a little overbearing about it,

1025
01:15:18,219 --> 01:15:20,539
but I don't know if there's a person

1026
01:15:20,539 --> 01:15:22,179
who's ever gone to an exchange

1027
01:15:22,179 --> 01:15:23,859
and knew they were getting real Bitcoin

1028
01:15:23,859 --> 01:15:27,559
until after the fact it hit their address.

1029
01:15:29,239 --> 01:15:30,539
Would I be wrong about that?

1030
01:15:31,199 --> 01:15:32,499
I don't think people do now

1031
01:15:32,499 --> 01:15:34,699
because it's not real until it's in your address.

1032
01:15:35,219 --> 01:15:43,079
And with that, I started thinking, you know what, if there was an exchange that did this proof and charged me, I would pay.

1033
01:15:43,079 --> 01:15:54,959
I would pay a little more clearly if exchange did this proof for me and put it on my basically essentially put it on my put it on my address before releasing it or some.

1034
01:15:55,499 --> 01:15:59,639
You know, I'm talking to the man probably who can actually figure this out.

1035
01:15:59,699 --> 01:16:00,999
I actually had to do this easy.

1036
01:16:01,179 --> 01:16:03,339
You mean one step beyond like proof of reserves.

1037
01:16:03,339 --> 01:16:09,759
You're talking about the transaction in flight showing you that they have the Bitcoin and that they're going to send it to you.

1038
01:16:09,759 --> 01:16:12,019
And it's real that it's actually like that.

1039
01:16:12,119 --> 01:16:13,699
I don't have to Karen, the manager.

1040
01:16:14,019 --> 01:16:18,119
It's right about all of the things we've been talking about in the math podcast.

1041
01:16:18,119 --> 01:16:32,479
But what I'm trying to really point out and illustrate is that people have like these deep sovereignty stacks where they have their nodes and their AirGraph wallets and they have their practices and their OPSEC.

1042
01:16:32,959 --> 01:16:41,319
And all of these things are super important, yet to me it breaks down at the point of sale completely.

1043
01:16:41,879 --> 01:16:44,819
And there's a big missing – there's just a big missing there.

1044
01:16:45,299 --> 01:16:46,359
There's a lot of trust.

1045
01:16:46,359 --> 01:16:50,459
Like all, it's almost like all of our trust is going to the exchange when we buy, when

1046
01:16:50,459 --> 01:16:52,859
you buy the Bitcoin and there's almost no exceptions to this.

1047
01:16:54,399 --> 01:16:57,419
You know, I think peer to peer probably lends itself a little bit more.

1048
01:16:57,519 --> 01:16:59,299
Well, it makes more sense to trust a peer.

1049
01:16:59,499 --> 01:17:03,539
We'll just put it that way because you, you have some accountability and you, you know,

1050
01:17:03,539 --> 01:17:07,799
it makes a little more sense to place trust in somebody, you know, but also a peer can,

1051
01:17:08,279 --> 01:17:10,019
you know, there's the hodl hodl type stuff.

1052
01:17:10,119 --> 01:17:12,099
There's the multi-sig escrow stuff you could do.

1053
01:17:12,099 --> 01:17:19,979
So but when you – you know how many people go to an exchange and say like I want to – I'm ready to put $2 million into Bitcoin.

1054
01:17:20,199 --> 01:17:20,799
Let's do this.

1055
01:17:20,919 --> 01:17:25,119
And then just by the stroke of luck, it hits their address.

1056
01:17:25,599 --> 01:17:27,919
They didn't think about that gap, right?

1057
01:17:27,959 --> 01:17:36,699
They should have had a briefcase with a handcuff on it tied to their wrist before they handed money over, right?

1058
01:17:36,819 --> 01:17:37,299
Right, right.

1059
01:17:37,919 --> 01:17:40,219
So like to me, this is like a massive thing.

1060
01:17:40,219 --> 01:17:43,339
I'm actually going to be basing certain presentations around this concept.

1061
01:17:43,819 --> 01:17:45,419
And I think I'm going to get a lot of hate for it.

1062
01:17:45,699 --> 01:17:47,859
But I think it's super important to point this out.

1063
01:17:49,339 --> 01:17:54,679
Yeah, I think the way I would think about it, the leader in this would be River with their proof of reserves.

1064
01:17:56,039 --> 01:18:00,119
They publicly say we hold 10,000 Bitcoin in custody of customers.

1065
01:18:00,119 --> 01:18:07,699
And they actually do this really cool Merkle tree proof that shows that for every liability on their books, there is Bitcoin in an address.

1066
01:18:08,259 --> 01:18:09,979
We need to do an episode on this.

1067
01:18:09,979 --> 01:18:21,019
And I think it's important because it's, again, on general proof of reserves and also like what River does and where – again, so they close maybe a portion of the gap.

1068
01:18:21,479 --> 01:18:24,939
So where does it end and where do we – where does the trust happen?

1069
01:18:25,479 --> 01:18:25,999
Yeah, we can do that.

1070
01:18:26,359 --> 01:18:29,739
River also fully open-sourced the code that does their proof of reserves.

1071
01:18:29,959 --> 01:18:31,959
They forked another open-source code base.

1072
01:18:31,959 --> 01:18:38,319
And that's kind of the point Alex always makes is that if you're in exchange, this isn't rocket science.

1073
01:18:38,439 --> 01:18:39,399
It's not super complicated.

1074
01:18:39,979 --> 01:18:45,979
You just need to have a map of your liabilities and tie it to a Bitcoin address or addresses to show that it's all matched.

1075
01:18:47,199 --> 01:18:51,439
So pre-images are a great solution to the problem.

1076
01:18:51,539 --> 01:18:57,459
And like I said, as a buyer, I would easily pay a little bit more to know, right?

1077
01:18:57,539 --> 01:19:01,039
Especially if the standard is good luck, buddy.

1078
01:19:01,799 --> 01:19:02,059
Right.

1079
01:19:03,199 --> 01:19:04,019
Very much.

1080
01:19:04,279 --> 01:19:05,059
Very, very much.

1081
01:19:05,499 --> 01:19:05,639
Yeah.

1082
01:19:06,099 --> 01:19:08,039
I think we easily could do that.

1083
01:19:08,039 --> 01:19:14,519
on uh we also do a little bit of merkle trees just sneak sit in there although it's not super

1084
01:19:14,519 --> 01:19:21,219
math heavy more just comp side but no i love i i have a i have at least one full discussion on

1085
01:19:21,219 --> 01:19:29,319
merkle trees in me excellent i think we can do that at least one all right cool anything else

1086
01:19:29,319 --> 01:19:34,279
you want to close it out i think we've closed it out for the day we we got through the residuals

1087
01:19:34,279 --> 01:19:36,279
and proving that something's a perfect square.

1088
01:19:37,839 --> 01:19:45,419
I think we've done a really good job on squeezing this lemon through this section.

1089
01:19:45,519 --> 01:19:48,919
I'm not sure what more we have, but you and I can take it offline

1090
01:19:48,919 --> 01:19:50,459
and figure out where to go next from here.

1091
01:19:51,119 --> 01:19:53,119
Yeah, it's a good analogy, squeezing a lemon.

1092
01:19:53,419 --> 01:19:56,139
Yeah, you got any cuts on your hands, it's going to hurt.

1093
01:19:56,599 --> 01:19:59,499
But then you're going to take a little ice, a little sugar, make some lemonade.

1094
01:19:59,659 --> 01:20:00,359
It's all going to be better.

1095
01:20:00,359 --> 01:20:08,019
So the next phase of I think what we'll be doing is it looks like we get three types of podcast episodes.

1096
01:20:08,139 --> 01:20:10,399
We got the ones that I just kind of do myself.

1097
01:20:11,099 --> 01:20:11,199
Yeah.

1098
01:20:11,379 --> 01:20:19,479
And we got the ones then the two types that you and I will be doing is one will be just be continuing on here.

1099
01:20:19,619 --> 01:20:23,319
But more now in the more focused on Bitcoin.

1100
01:20:24,199 --> 01:20:24,679
Right.

1101
01:20:24,799 --> 01:20:25,919
More focused on now.

1102
01:20:25,919 --> 01:20:26,339
what do we,

1103
01:20:26,779 --> 01:20:27,419
we're going to,

1104
01:20:27,459 --> 01:20:32,919
we're going to assume now you guys have beefed up on all this arithmetic and

1105
01:20:32,919 --> 01:20:34,579
you know why we did it.

1106
01:20:34,619 --> 01:20:36,979
Now we're going to talk kind of about signatures,

1107
01:20:38,319 --> 01:20:39,579
elliptical transactions,

1108
01:20:40,659 --> 01:20:41,239
stuff like that.

1109
01:20:41,719 --> 01:20:42,699
And then the other one is,

1110
01:20:42,819 --> 01:20:43,099
you know,

1111
01:20:43,179 --> 01:20:47,539
we do intend to have some friends join us.

1112
01:20:47,959 --> 01:20:48,219
Yes.

1113
01:20:48,819 --> 01:20:52,459
As soon as maybe even end of next week,

1114
01:20:53,199 --> 01:20:54,919
we'll see if we can get some people.

1115
01:20:55,919 --> 01:20:59,879
And then also through April, I think we're going to get ourselves busy.

1116
01:21:00,559 --> 01:21:06,599
I think in April, I have three straight scheduled weeks where I'm going to be in the same room with you.

1117
01:21:08,579 --> 01:21:09,019
Interesting.

1118
01:21:09,259 --> 01:21:09,779
You're right.

1119
01:21:10,239 --> 01:21:12,879
First week, second week, and then the third week, oh, Vegas?

1120
01:21:13,639 --> 01:21:13,899
Yeah.

1121
01:21:14,459 --> 01:21:15,739
Man, it could be wild.

1122
01:21:16,299 --> 01:21:16,919
It could be wild.

1123
01:21:16,919 --> 01:21:22,459
So I got my roadcaster ready to hit the road, ready to do rips.

1124
01:21:22,999 --> 01:21:23,759
Let's do it.

1125
01:21:23,759 --> 01:21:26,339
And then, you know, we're going to be in the same room.

1126
01:21:27,139 --> 01:21:33,439
Now, you know, we're going to be in the vicinity of, I think, some people that could really be good now.

1127
01:21:33,499 --> 01:21:36,659
I think Opinx is going to be a great one for us to do some breakout sessions.

1128
01:21:37,239 --> 01:21:37,439
Yep.

1129
01:21:37,459 --> 01:21:41,559
And I already put the application in with Charlie.

1130
01:21:42,019 --> 01:21:44,779
So, look, we'll be good to go there.

1131
01:21:45,199 --> 01:21:48,079
You know, the fact that we do audio only makes it very easy.

1132
01:21:48,199 --> 01:21:48,939
Very straightforward.

1133
01:21:49,319 --> 01:21:49,399
Yeah.

1134
01:21:49,599 --> 01:21:49,799
Yep.

1135
01:21:50,519 --> 01:21:50,839
All right.

1136
01:21:51,299 --> 01:21:51,619
Awesome.

1137
01:21:51,619 --> 01:21:51,759
Awesome.

1138
01:21:52,279 --> 01:21:52,699
Sounds good.

1139
01:21:52,699 --> 01:21:54,219
Guys, take your pencils out.

1140
01:21:54,459 --> 01:21:56,779
I'm looking for catchphrases until close to subject.

1141
01:21:58,179 --> 01:22:00,179
Like Tom Malongo, keep your stick on the ice.

1142
01:22:00,999 --> 01:22:02,779
Take your pencils out.

1143
01:22:05,419 --> 01:22:09,819
Beef up because we're about to level up here.

1144
01:22:10,159 --> 01:22:10,479
Okay, guys?

1145
01:22:10,599 --> 01:22:10,779
That's right.

1146
01:22:11,299 --> 01:22:11,619
All right.

1147
01:22:11,659 --> 01:22:11,919
Awesome.

1148
01:22:12,099 --> 01:22:12,339
See you.

1149
01:22:12,659 --> 01:22:12,959
See you.