1 00:00:00,820 --> 00:00:02,180 All right. Good morning. 2 00:00:03,240 --> 00:00:04,240 Good morning. 3 00:00:04,440 --> 00:00:07,360 Let's let everybody know what time it is because, you know, that's important here. 4 00:00:07,600 --> 00:00:10,440 But good morning, man. Have you said your good mornings? 5 00:00:10,820 --> 00:00:15,380 I just did. I just did. You got it on camera. You got it recorded. 6 00:00:16,000 --> 00:00:17,240 That's right. We do have it on camera. 7 00:00:18,460 --> 00:00:21,620 This is the Magic Internet Math Podcast, Episode 4. 8 00:00:21,620 --> 00:00:28,140 and I think this will probably be the second part two of what we called 9 00:00:28,140 --> 00:00:30,580 the elliptic curve cryptography study guide. 10 00:00:31,100 --> 00:00:31,200 Yeah. 11 00:00:31,980 --> 00:00:36,160 We started in last week talking about some clock math, 12 00:00:36,740 --> 00:00:39,520 which formerly known as modular arithmetic. 13 00:00:41,300 --> 00:00:42,360 Very powerful. 14 00:00:42,800 --> 00:00:45,900 Probably one of the things I'm the most excited about that has ever existed in 15 00:00:45,900 --> 00:00:50,740 the world is this ability to do modular arithmetic just for the hell of it. 16 00:00:51,620 --> 00:00:52,660 Just because we can. 17 00:00:53,860 --> 00:00:54,240 Yeah. 18 00:00:54,460 --> 00:00:57,160 And then like, what has that given us? 19 00:00:57,320 --> 00:01:01,480 You know, not to think, boy, this thing out too much, but I just, it's such a primitive, 20 00:01:01,680 --> 00:01:04,060 it's such a powerful primitive and it's going to fit. 21 00:01:04,200 --> 00:01:08,520 It's going to feature very heavily now and what we're going to talk about today. 22 00:01:09,000 --> 00:01:10,200 Yeah, I'd say so. 23 00:01:10,200 --> 00:01:15,940 I think what we did last time is we kind of at a very high level connected the core concepts 24 00:01:15,940 --> 00:01:19,260 of these math pieces that actually relate to Bitcoin. 25 00:01:19,260 --> 00:01:33,880 And we kind of at a very high level, like a survey level, just quick pass through, talked about all of these mathematical interdependencies that make the system of Bitcoin security, which is the public private key pair. 26 00:01:34,580 --> 00:01:38,400 Right. And so we're going to slow it down a little bit today. 27 00:01:38,840 --> 00:01:43,560 We're going to go a little bit more into some of the interesting mathematical properties that yield that. 28 00:01:43,560 --> 00:01:51,280 and if I would recommend starting it last episode even if you don't grok 100% of every single beat 29 00:01:51,280 --> 00:01:55,840 you'll see kind of the big picture vision of what we're trying to connect and ground these concepts 30 00:01:55,840 --> 00:02:01,660 for which is the securing of bitcoin getting a public agree being able to have a private key 31 00:02:01,660 --> 00:02:08,260 being able to reveal information that you have a private key but no one else knows that private key 32 00:02:08,260 --> 00:02:12,540 and the mathematical properties that we use with this a little bit of curve we're going to slow it 33 00:02:12,540 --> 00:02:16,640 down a little bit more today and talk about a lot of those like smaller building blocks. 34 00:02:17,780 --> 00:02:22,400 Yeah. And I like, I mean, high level is a great way to put it. And it's just that if you don't, 35 00:02:22,420 --> 00:02:28,900 if you don't like, you don't see the depth of it, what's the meme with the iceberg where all 36 00:02:28,900 --> 00:02:33,200 you see is the tip of the iceberg. So like at the tip of the iceberg, you see Bitcoin working, 37 00:02:33,480 --> 00:02:39,000 but like, I feel like in some ways we're the bearer of bad news by pointing out, 38 00:02:39,000 --> 00:02:45,660 we're starting to talk about the bottom of the iceberg and just and say hey trust us for now 39 00:02:45,660 --> 00:02:53,360 that it's connected stick with us right and um i think it's important to see it even though we you 40 00:02:53,360 --> 00:02:58,640 know don't kill the messenger we are that's we're we're trying to create a rich conversation here 41 00:02:58,640 --> 00:03:03,620 for an understanding of something we all really care about very deeply yeah i would say last 42 00:03:03,620 --> 00:03:07,700 episode we sketched out the shape of the entire iceberg and now we're going to start coloring 43 00:03:07,700 --> 00:03:10,980 and all of the little details at the bottom and work our way up. 44 00:03:12,240 --> 00:03:12,640 Yeah. 45 00:03:12,980 --> 00:03:13,580 Yeah, it's good. 46 00:03:13,740 --> 00:03:17,020 So I can't overemphasize. 47 00:03:17,480 --> 00:03:20,940 Oh, the thing I forgot to say was go look at the study guide in the show notes. 48 00:03:20,940 --> 00:03:24,220 That was like the one mental prompt I gave myself before I just said what I said. 49 00:03:24,480 --> 00:03:25,160 I forgot. 50 00:03:25,820 --> 00:03:34,100 Go look at the study guide because what we're doing now is I think we have a little bit of a three-way attack on this. 51 00:03:34,200 --> 00:03:37,520 We have this podcast where we awkwardly explain these things. 52 00:03:37,520 --> 00:03:40,760 But mainly I think what we're doing in the podcast is we're showing you what's important. 53 00:03:41,400 --> 00:03:53,000 And then when you want to go and really grok it, we have a study guide and I have a YouTube series now that is available in the show notes. 54 00:03:54,020 --> 00:04:02,140 So like I think if you want to – I mean obviously I don't think anybody expects to master the math of anything by listening to a podcast, right? 55 00:04:02,140 --> 00:04:10,140 And I think my experience of doing this the first time with Gary, shout out, is it's hard. 56 00:04:10,280 --> 00:04:10,960 It's like frustrating. 57 00:04:11,100 --> 00:04:15,800 A lot of people said, man, it's so hard to sit through this podcast. 58 00:04:16,020 --> 00:04:24,200 So like what I want you guys, I think it's less difficult to know there are supporting, other supporting tools now. 59 00:04:25,200 --> 00:04:32,000 Yeah, I would say every person learns differently and kind of what's effective for them to be able to unlock concepts. 60 00:04:32,140 --> 00:04:36,520 me personally what I would do is I would read the study guide then listen to the podcast 61 00:04:36,520 --> 00:04:40,360 because when you read the study guide and you're able to throw your hands up and say 62 00:04:40,360 --> 00:04:44,360 this is too complicated for me I don't get this when we go 63 00:04:44,360 --> 00:04:48,200 re-explain it on the show that may be able to provide you different mental models to be able to 64 00:04:48,200 --> 00:04:52,160 kind of get across the chasm and then additionally drop us a 65 00:04:52,160 --> 00:04:55,680 comment on fountain if there's a part that you want us to go deeper into 66 00:04:55,680 --> 00:05:00,300 absolutely like if you're listening along you're flipping through the study 67 00:05:00,300 --> 00:05:03,160 guide you're listening to the podcast and there's things that you just still don't quite get 68 00:05:03,160 --> 00:05:08,000 like just drop a line to us and it could be like a little bit of an asynchronous uh study group 69 00:05:08,000 --> 00:05:14,420 that we all go through on this journey together oh yeah i should also mention before we jump in 70 00:05:14,420 --> 00:05:19,880 too quickly is that um i mean this is going off a lot of people but like there are 71 00:05:19,880 --> 00:05:27,160 there is a membership available at the magic internet math academy and um right now what 72 00:05:27,160 --> 00:05:31,120 you see is everything that is there is freely available. So that's just the tip of the iceberg. 73 00:05:31,380 --> 00:05:36,200 So I haven't yet identified what's below it, but there will definitely be like subscriber only 74 00:05:36,200 --> 00:05:43,040 content. And there's a few people, it's like $2 a month. Um, if you want to do that, um, for a 75 00:05:43,040 --> 00:05:48,040 very limited time right now, I'm accepting lifetime memberships, um, in Bitcoin only. 76 00:05:49,180 --> 00:05:54,880 And so we have four members so far, shout out to those guys. And, um, so let's get to it. 77 00:05:54,880 --> 00:05:55,680 Make sense? 78 00:05:57,480 --> 00:05:58,720 That sounds good to me. 79 00:05:59,000 --> 00:05:59,540 All right. 80 00:06:00,020 --> 00:06:04,700 So I think what we want to do is start with Chapter 2 of the study guide. 81 00:06:05,280 --> 00:06:05,480 Okay? 82 00:06:05,900 --> 00:06:12,040 And so anybody who already looked at study guide is about to pay off by knowing what's in Chapter 2. 83 00:06:13,800 --> 00:06:19,340 And we did – you know, we hit this last week, but did – I mean, this repetition is everything. 84 00:06:19,340 --> 00:06:24,560 And so we're going to repeat – what Chapter 2 is called is the inverse problem. 85 00:06:24,880 --> 00:06:43,460 And, you know, inverse is, it's like of real importance to know that your private key, which is a point on the elliptic curve, right, gets multiplied by some generator that is another point on the elliptic curve that the public sees. 86 00:06:44,320 --> 00:06:52,440 It's of utmost importance that you know you can invert that public key back into your private key. 87 00:06:53,640 --> 00:06:54,200 Right? 88 00:06:54,880 --> 00:07:01,320 Like that is of, imagine if for some reason you were in a number system where that was 89 00:07:01,320 --> 00:07:05,600 not possible and it can only happen by dividing by zero, which anybody who's ever tried to 90 00:07:05,600 --> 00:07:11,840 code anything knows how easy that can happen at a given point in time if they're not severely 91 00:07:11,840 --> 00:07:14,400 restricted by the number system they're working with. 92 00:07:15,540 --> 00:07:20,060 So what we kind of talked about with modular arithmetic a little bit and then gotten to 93 00:07:20,060 --> 00:07:28,420 groups right we went we went very we took a very large leap across stones and went to groups because 94 00:07:28,420 --> 00:07:34,000 what groups and fields do is ensure and what they really do is ensure the existence of an inverse 95 00:07:34,000 --> 00:07:40,380 that's one of the things they do and when we're dealing with things like hash functions where 96 00:07:40,380 --> 00:07:49,320 you know you just have to knowing the math allows you to know the inverse exists without really 97 00:07:49,320 --> 00:07:54,540 being able to do the math with a pen and paper or even with a computer. Right. So like we're kind 98 00:07:54,540 --> 00:08:00,520 of in this realm of like, you have to, you have to, there's certain things now you do have to 99 00:08:00,520 --> 00:08:06,980 trust and you have to trust an inverse exists with a hash function, even though you can't really 100 00:08:06,980 --> 00:08:11,440 do it. Like, you know, you know, if you're into real numbers and you have a number five, 101 00:08:11,740 --> 00:08:16,040 you know, one fifth is the inverse of that. You can handle that and your brain can handle that. 102 00:08:16,040 --> 00:08:20,740 If you're in a certain number system 103 00:08:20,740 --> 00:08:23,040 That's, you know 104 00:08:23,040 --> 00:08:25,320 Most number systems and most numbers 105 00:08:25,320 --> 00:08:27,300 Your computer can handle, right? 106 00:08:27,360 --> 00:08:28,340 But the whole point of this 107 00:08:28,340 --> 00:08:30,580 The whole point is your computer can't handle it 108 00:08:30,580 --> 00:08:31,580 And no computer can 109 00:08:31,580 --> 00:08:33,060 No group of computers can 110 00:08:33,060 --> 00:08:36,400 In fact, as long as you're on this earth 111 00:08:36,400 --> 00:08:39,100 And subject to the constraints of energy 112 00:08:39,100 --> 00:08:42,840 And the current state of, you know 113 00:08:42,840 --> 00:08:44,160 Algorithms and all that 114 00:08:44,160 --> 00:08:50,460 there's nothing you can do to actually sit and prove that your inverse exists. You have to trust 115 00:08:50,460 --> 00:08:55,480 the math on this one, but that's why we're here. That's what I'm motivating at the moment is, 116 00:08:55,480 --> 00:09:02,420 how do you know you have an inverse? And this is, again, as just one high-level concept we 117 00:09:02,420 --> 00:09:09,400 talked about on the last podcast, is that the reason why you basically have to remove 118 00:09:09,400 --> 00:09:14,180 a brute force like always checking and you have to go to an abstract theoretical proof 119 00:09:14,180 --> 00:09:20,120 is that the number of points on this curve outnumbers the atoms that exist in the universe 120 00:09:20,120 --> 00:09:27,720 and since it requires energy computation and time none of this stuff is free even with moore's law 121 00:09:27,720 --> 00:09:34,140 and like doubling every 18 months the ability for us to compute things this is so many orders of 122 00:09:34,140 --> 00:09:40,220 magnitude outside of the scope of what could be kind of infinite like being able to definitively 123 00:09:40,220 --> 00:09:44,560 every single point on the curve proving it has an inverse through brute force is impossible you're 124 00:09:44,560 --> 00:09:49,620 you're looking at the heat death of the universe even with quantum times over which i mean i feel 125 00:09:49,620 --> 00:09:56,160 like we'll mention the word but like even it's true but so like the thing that's a good point 126 00:09:56,160 --> 00:09:59,880 with quantum is that quantum is not going to be able to definitively show every single point on a 127 00:09:59,880 --> 00:10:05,740 curve, it can find an individual point of a public key and find that one inverse to the 128 00:10:05,740 --> 00:10:06,300 private key. 129 00:10:06,660 --> 00:10:11,020 But it's not going to be able, theoretically, but it's not going to be able to brute force 130 00:10:11,020 --> 00:10:12,940 and show the universe of all possible keys. 131 00:10:13,620 --> 00:10:13,740 Yeah. 132 00:10:13,820 --> 00:10:18,560 So what I kind of want to, before we get into it, it's like when you go to the Bitcoin store 133 00:10:18,560 --> 00:10:21,640 and say, I am ready to buy Bitcoin, right? 134 00:10:23,300 --> 00:10:28,480 Theoretically, as a customer, you already, I don't know how many people own Bitcoin, 135 00:10:28,480 --> 00:10:39,080 But every single one of them should have gone to the counter and asked to see the manager and say, I need you to – before I buy this Bitcoin, can you just show me that I have – that an inverse exists to my private key? 136 00:10:40,760 --> 00:10:44,240 Theoretically, that is the thing you should be doing at the store. 137 00:10:44,940 --> 00:10:45,040 Yeah. 138 00:10:45,780 --> 00:10:46,120 OK. 139 00:10:46,420 --> 00:10:48,640 And that's called verify. 140 00:10:49,280 --> 00:10:49,500 OK. 141 00:10:49,560 --> 00:10:51,240 That's the thing that we call verify. 142 00:10:51,380 --> 00:10:56,980 When we say don't trust verify, I think in my opinion, it goes that – it should go that deep. 143 00:10:57,120 --> 00:10:58,300 Like how do you – 144 00:10:58,300 --> 00:11:03,660 Well, it has to because otherwise, how do you know if you generate a Bitcoin address that there's actually a corresponding private key? 145 00:11:04,240 --> 00:11:04,880 That's right. 146 00:11:04,880 --> 00:11:05,960 Until you check it yourself manually. 147 00:11:06,200 --> 00:11:06,600 That's right. 148 00:11:06,680 --> 00:11:07,140 You can't. 149 00:11:07,140 --> 00:11:13,680 So this evergreen idea of an inverse is the way this happens. 150 00:11:13,680 --> 00:11:32,100 Right. It's the way a person motivated to actually maybe who has that a good natural barrier for not wanting to trust his life savings to a bunch of zeros and ones that may or that may not pan out or be spendable. 151 00:11:32,360 --> 00:11:40,080 Right. This is a big this concept of inverse is now that is one of the things that can get you over a very big hump. 152 00:11:40,080 --> 00:11:43,500 I'm really saying you should be asking this question. 153 00:11:43,660 --> 00:11:45,120 People should be asking this question. 154 00:11:45,220 --> 00:11:48,760 People should not tolerate owning Bitcoin and not asking this question. 155 00:11:48,920 --> 00:11:53,220 So what I would love to do is we should explain this inverse property. 156 00:11:53,820 --> 00:11:58,980 And then I'm going to tie it into LibSecP and how the library actually works. 157 00:11:59,560 --> 00:12:01,600 Because it's a little interesting thing we'll get to. 158 00:12:01,640 --> 00:12:03,300 But I want to lay out the core math first. 159 00:12:03,920 --> 00:12:04,360 So, okay. 160 00:12:04,500 --> 00:12:09,800 So real quick, before we talk about an inverse, we're going to talk about something called the identity. 161 00:12:10,080 --> 00:12:36,420 So this is very abstract. However, there's a reason method for it. So like, you know, an identity under addition, I think people can handle is the number zero, like that's the identity, meaning like zero plus anything is that number. Right? So like, when you talk about an identity, you want to know that you it's the thing. This sounds so like pedantic, but it's really, really important. 162 00:12:36,420 --> 00:12:43,840 it's like what can i how do i operate on a number and not change it at all and the answer is you use 163 00:12:43,840 --> 00:12:50,860 the identity you you operate on the identity and that's really important and the identity changes 164 00:12:50,860 --> 00:12:56,280 based on what the problem set is so and the operation and the operation so i think the other 165 00:12:56,280 --> 00:13:02,680 one i think most people would be able to think through is with instead of addition multiplication 166 00:13:02,680 --> 00:13:05,980 And I want everyone to take a half second and think before we say the answer. 167 00:13:06,360 --> 00:13:10,780 What do you do in multiplication to get the same number after you multiply it? 168 00:13:11,300 --> 00:13:11,960 That's right. 169 00:13:12,140 --> 00:13:13,940 The number one, right? 170 00:13:14,040 --> 00:13:15,840 Two times one is one, right? 171 00:13:15,880 --> 00:13:20,860 So you can see that the identity is not a constant across all methods and all perspectives. 172 00:13:21,400 --> 00:13:26,720 But the identity as a concept, though, existing is really important for the stuff we're going to be talking about. 173 00:13:27,440 --> 00:13:27,880 Yes. 174 00:13:27,880 --> 00:13:31,280 And this is why we talk about groups, rings, and fields. 175 00:13:31,800 --> 00:13:37,080 Because a group gives you an identity for an operation. 176 00:13:37,220 --> 00:13:41,460 It doesn't tell you what the operation is, so it doesn't tell you what the identity is. 177 00:13:42,080 --> 00:13:49,940 However, a group is just a primitive to understand this concept that you have, again, closure, identity. 178 00:13:50,760 --> 00:13:53,220 And if you have identity, you have an inverse. 179 00:13:53,220 --> 00:13:59,060 because the inverse is the element that gets you back to the identity. 180 00:13:59,280 --> 00:14:02,620 So when we talked about addition, in the integers, let's just say, 181 00:14:02,720 --> 00:14:06,340 if you have the number 5 and the identity is 0, 182 00:14:06,520 --> 00:14:09,960 because we said 0 is the identity for addition, 183 00:14:11,340 --> 00:14:15,700 then the number that gets you back to 0 is negative 5, 184 00:14:16,180 --> 00:14:18,160 which happens not to exist. 185 00:14:18,300 --> 00:14:20,020 Well, no, sorry, it does exist in the integers. 186 00:14:20,440 --> 00:14:21,020 It does. 187 00:14:21,020 --> 00:14:32,640 Yes. So under addition, you're good there, right? So five plus negative five is zero. So therefore, negative five is your inverse, your additive inverse, your inverse under the operation of addition. 188 00:14:32,640 --> 00:14:41,500 And to take it to multiplication, if I have the number two, what can be combined with two to always get back to the inverse? 189 00:14:42,140 --> 00:14:46,460 Well, what is the inverse in that case that gets you back to the identity of one? 190 00:14:46,980 --> 00:14:48,780 It's one half, right? 191 00:14:48,820 --> 00:14:49,660 One over two. 192 00:14:49,940 --> 00:14:52,620 So two times one over two equals one, which is the identity. 193 00:14:53,120 --> 00:15:02,100 So that's another example about how you can get the identity and you can compute it depending on the field, like what you're doing as the operation and changes. 194 00:15:02,100 --> 00:15:07,040 But there is an inverse that exists for all the real numbers, and it's one over that. 195 00:15:07,680 --> 00:15:12,240 And we're so confident in our – the human race is so confident about this. 196 00:15:12,320 --> 00:15:16,060 We've given both of those things names because they're so general. 197 00:15:16,640 --> 00:15:22,360 We call the additive inverse like an opposite or a negative, a negation. 198 00:15:22,820 --> 00:15:26,220 And we call the multiplicative inverse a reciprocal. 199 00:15:26,620 --> 00:15:30,620 And I'm saying this because I'm pretty sure everybody knows what these words are, right? 200 00:15:30,620 --> 00:15:35,940 But now I want those words to be associated. 201 00:15:36,160 --> 00:15:39,060 It's easier to associate this concept now. 202 00:15:39,640 --> 00:15:39,820 Yes. 203 00:15:39,940 --> 00:15:45,360 The reciprocal is the multiplicative inverse, and the negative is the additive inverse. 204 00:15:45,520 --> 00:15:48,000 And why do we focus on addition and multiplication here? 205 00:15:48,560 --> 00:15:50,640 And why do we talk about rings and fields? 206 00:15:50,760 --> 00:15:56,500 It's because in a ring and a field now, you have both operations, addition and multiplication. 207 00:15:56,500 --> 00:16:04,560 and so um in a field a field is like a perfect version of a ring where you have an inverse for 208 00:16:04,560 --> 00:16:10,340 both right you have an additive inverse so i don't want to get to a field here but like i just want 209 00:16:10,340 --> 00:16:17,360 the reason we talk about fields is a field is like um a statue and the ring is more like the block 210 00:16:17,360 --> 00:16:23,900 of clay that we know can become a statue yeah right and this and the reason why we're doing 211 00:16:23,900 --> 00:16:30,900 this is because remember what we talked about on the last episode is that we're not dealing with 212 00:16:30,900 --> 00:16:38,540 just the number line and all the real numbers there's a very particular set in order of the 213 00:16:38,540 --> 00:16:44,040 points that we deal with in the bitcoin elliptic curve which was y cubed equals x squared plus seven 214 00:16:44,040 --> 00:16:49,960 that's where this all works and we're not looking at everything we're looking at things within that 215 00:16:49,960 --> 00:16:54,160 subset and the relationship of how that all works, which we briefly touched upon last 216 00:16:54,160 --> 00:16:55,960 episode, creates a field. 217 00:16:56,420 --> 00:17:00,640 And that's where the modulo arithmetic and everything starts coming into play, is that 218 00:17:00,640 --> 00:17:03,560 we're not just doing negative infinity to positive infinity. 219 00:17:03,820 --> 00:17:06,220 There's a very particular set of numbers that we're working on. 220 00:17:07,020 --> 00:17:07,380 That's right. 221 00:17:07,380 --> 00:17:15,580 I mean, imagine being Satoshi, right, and worrying about, worrying in 100 years that 222 00:17:15,580 --> 00:17:23,620 Someone's going to try to spend, you know, do a spend and they're going to run into some divide by zero or infinity thing. 223 00:17:23,780 --> 00:17:27,220 I mean, it's like it had to be eliminated. 224 00:17:28,440 --> 00:17:30,700 Right. It had to be eliminated on day zero. 225 00:17:31,320 --> 00:17:37,520 So we have we use a number system, not the integers or the reals. 226 00:17:37,580 --> 00:17:40,920 We use a number system called a finite field modulo prime. 227 00:17:40,920 --> 00:17:47,640 And I know that's confusing a little bit, but if you go back to last episode, we talked a good amount about it. 228 00:17:47,720 --> 00:17:56,300 And it's really just a compact system that operates under the operations of addition and multiplication. 229 00:17:57,840 --> 00:18:10,040 And if we go back to the modulo arithmetic, it just so happens that modulo arithmetic follows the same rules as regular arithmetic almost across the board. 230 00:18:10,040 --> 00:18:18,260 And so it's a very nice, convenient way of now being able to do math, this type of math, in a way that ensures this structure. 231 00:18:18,820 --> 00:18:23,920 The structure, again, meaning closure, identity inverse under addition and multiplication. 232 00:18:24,320 --> 00:18:28,640 This is just – and the reason – and here's the thing I can't reiterate enough. 233 00:18:29,100 --> 00:18:31,440 Why do we even talk about this structure? 234 00:18:31,700 --> 00:18:34,560 It's like – I know everyone is like, why the hell are you talking about this structure? 235 00:18:34,620 --> 00:18:36,820 It's so alienating and I'm already confused. 236 00:18:36,820 --> 00:18:43,840 And the answer to this is that you can't do this math on a pen and paper or with a computer or with all the computers in the world. 237 00:18:43,920 --> 00:18:53,340 So you have to be able to trust that the math is doable no matter what and no matter what the situation is. 238 00:18:53,340 --> 00:19:03,640 Yeah, and I'd even add we're just laying out the axiomatic baseline principles of where you get to the point to be able to understand the system works as it is designed. 239 00:19:03,740 --> 00:19:04,900 And it's less axiomatic. 240 00:19:04,900 --> 00:19:09,420 Like each one of these has a mathematical proof and a reasoning behind it. 241 00:19:09,500 --> 00:19:14,020 It is not – it is because we say so, which is what an axiom is, right? 242 00:19:14,040 --> 00:19:16,240 It's like just a definitive like base truth. 243 00:19:16,900 --> 00:19:25,000 But we're trying to lay out all of these theoretical concepts so that you're able to know, not trust how this all actually works. 244 00:19:25,000 --> 00:19:25,340 That's right. 245 00:19:25,340 --> 00:19:31,260 It's all about knowing without even – without the ability to actually do it. 246 00:19:31,340 --> 00:19:34,760 Normally the way I know how arithmetic works is I sit and do it. 247 00:19:34,900 --> 00:19:49,620 Right. I know three plus three equals six. I can sit there. I can count my fingers. I can, you know, I can know that. The things we're talking about are in a world where you just can't know by the same type of ways that we know things. 248 00:19:49,620 --> 00:19:55,040 and it's not even like we can even trust our computers to do it either. 249 00:19:55,040 --> 00:20:00,400 You have to know that the structure, you just have to understand the structure 250 00:20:00,400 --> 00:20:05,540 and say, you know what, the structure preserves these important things. 251 00:20:05,660 --> 00:20:08,740 It discards everything else, right? 252 00:20:08,840 --> 00:20:14,960 I don't need to know anything other than that these operations of multiplication 253 00:20:14,960 --> 00:20:18,300 and addition are preserved and that these properties of the group are preserved. 254 00:20:18,300 --> 00:20:26,060 That's all I need to know to be able to essentially have knowledge of this kind of unverifiable thing. 255 00:20:26,900 --> 00:20:27,000 Right. 256 00:20:27,500 --> 00:20:34,840 So with all of that context laid out, let's talk about these multiplicative inverses and how they work with Bitcoin. 257 00:20:35,980 --> 00:20:36,540 Yeah. 258 00:20:36,540 --> 00:20:44,800 Before we talk about Bitcoin, let's just really in a finite field. 259 00:20:44,800 --> 00:20:50,620 you know like i'll talk about a finite field maybe modulo seven and one like modulo six 260 00:20:50,620 --> 00:20:54,580 right like so like that's a good place to start just keep it super simple 261 00:20:54,580 --> 00:21:01,080 base abstraction of the concept in uh a domain space which is very easy to understand 262 00:21:01,080 --> 00:21:06,940 that's right so like and once again i should have all of this written out and prepared out 263 00:21:06,940 --> 00:21:12,140 prepared out so i'm not doing this in my head however let's just look at a finite field modulo 264 00:21:12,140 --> 00:21:16,320 7, which we said would consist of the numbers 0 through 6. 265 00:21:18,160 --> 00:21:24,720 And the reason why is if you got to the number 7, 7 divided by 7 is 1. 266 00:21:25,040 --> 00:21:29,420 Actually, it doesn't include 0 either, so my bad on that. 267 00:21:30,380 --> 00:21:30,680 There you go. 268 00:21:30,680 --> 00:21:33,460 But 1, you know, 1, 2, it's basically what it does. 269 00:21:33,680 --> 00:21:34,240 Remainder, sorry. 270 00:21:34,380 --> 00:21:35,160 The remainder is 0. 271 00:21:35,220 --> 00:21:37,660 If you do 7, the remainder is 0, so modulo is 0. 272 00:21:38,060 --> 00:21:38,360 That's right. 273 00:21:38,360 --> 00:21:44,440 Now, I do have a video on this that's buried in my YouTube site that I did last year. 274 00:21:44,540 --> 00:21:50,820 But basically, what you have, the finite field consists of all of the numbers that are relatively prime to your modulus. 275 00:21:51,340 --> 00:21:57,720 So any number relatively prime to 7, which if that number is prime, it's all of the numbers before 7. 276 00:21:57,900 --> 00:22:03,180 So 1, 2, 3, 4, 5, and 6 are what you call relatively prime because they don't share a common factor. 277 00:22:03,180 --> 00:22:18,740 Whereas if the modulus was 6, then you don't have that finite field because 2 is – you have 2, 4 and 4, which are not relatively prime. 278 00:22:19,640 --> 00:22:25,460 So these numbers don't have inverses in the modulus 6 because I would say 2 times 3 is 0. 279 00:22:26,220 --> 00:22:28,460 So that's unacceptable in a finite field. 280 00:22:28,660 --> 00:22:29,960 So now you have no inverse. 281 00:22:30,480 --> 00:22:33,600 What you're looking for is two numbers that multiply to one. 282 00:22:33,740 --> 00:22:34,620 They're called associates. 283 00:22:35,240 --> 00:22:38,760 But that's what you're really looking for is that you have a number in your number system. 284 00:22:39,440 --> 00:22:44,380 And you have two numbers in your number system that can multiply to one. 285 00:22:44,380 --> 00:22:52,760 And just to drive that home, why you don't want to be able to get to zero is that if you get to zero and multiple points get to zero, you don't know what your original input is anymore. 286 00:22:53,360 --> 00:22:54,140 You can't get there. 287 00:22:54,420 --> 00:22:54,740 That's right. 288 00:22:54,740 --> 00:23:09,656 You cannot guarantee now that the random number you created can be traced back Right You have a collision in another way You have two different you could have multiple numbers that result to the same point on the curve 289 00:23:09,756 --> 00:23:10,716 which doesn't work. 290 00:23:12,556 --> 00:23:14,516 Oh, you know what I'm being reminded? 291 00:23:14,656 --> 00:23:17,716 So I'm creating a basic algebra series, 292 00:23:18,136 --> 00:23:19,216 like high school algebra series. 293 00:23:19,376 --> 00:23:20,376 And one of the, 294 00:23:21,476 --> 00:23:22,916 I think when we get into inverses, 295 00:23:23,296 --> 00:23:26,976 there's this story in mythology 296 00:23:26,976 --> 00:23:30,956 about somebody who had to go through a cave 297 00:23:30,956 --> 00:23:33,176 and fight all these monsters. 298 00:23:34,036 --> 00:23:38,796 And the gods gave different weapons to different people. 299 00:23:39,296 --> 00:23:40,896 And the person who traversed the cave 300 00:23:40,896 --> 00:23:42,696 is the one the gods gave a thread. 301 00:23:43,676 --> 00:23:45,576 Because everyone who went in this cave 302 00:23:45,576 --> 00:23:46,576 couldn't find their way out, 303 00:23:46,716 --> 00:23:49,016 except the person whose weapon was a thread 304 00:23:49,016 --> 00:23:51,816 was able to find his way back out of the cave 305 00:23:51,816 --> 00:23:56,436 after being able to defeat all the monsters in the cave. 306 00:23:56,436 --> 00:23:58,356 And that's like really what this is. 307 00:23:58,716 --> 00:24:09,696 This is you holding a thread in a dark cave and accomplishing whatever the quest is and finding your way back out of the cave. 308 00:24:09,896 --> 00:24:14,836 And you want to – but you have to – the thread is the thing that tells you you can find your way out. 309 00:24:15,456 --> 00:24:18,036 You're not going to be lost at sea, right? 310 00:24:18,896 --> 00:24:19,376 Totally. 311 00:24:19,636 --> 00:24:24,416 And I think in the interest of building that thread, let's keep it to the simplest example with Modulo 7. 312 00:24:24,416 --> 00:24:25,996 let's say okay 313 00:24:25,996 --> 00:24:28,036 1 modulo 7 314 00:24:28,036 --> 00:24:29,316 is 1 315 00:24:29,316 --> 00:24:30,576 that's my identity 316 00:24:30,576 --> 00:24:33,276 2 modulo 7 317 00:24:33,276 --> 00:24:34,736 is 2 318 00:24:34,736 --> 00:24:37,216 and you can go all the way up to 6 319 00:24:37,216 --> 00:24:39,896 is 320 00:24:39,896 --> 00:24:40,936 you know 321 00:24:40,936 --> 00:24:43,456 6 right 322 00:24:43,456 --> 00:24:45,796 and like if you jump ahead and say 323 00:24:45,796 --> 00:24:47,316 okay what's 10 modulo 7 324 00:24:47,316 --> 00:24:49,996 well if you divide 10 by 7 325 00:24:49,996 --> 00:24:51,156 your remainder is 3 326 00:24:51,156 --> 00:24:53,496 so your remainder is now 3 327 00:24:53,496 --> 00:25:01,556 so let's talk you want to hit the you want let's identify the pair the associates that you know 328 00:25:01,556 --> 00:25:08,816 when you pair them up are inverses of each other in the in the finite field modulo seven so if we 329 00:25:08,816 --> 00:25:17,216 have two right the inverse is the thing that we multiply to two that gives us one which is four 330 00:25:17,216 --> 00:25:21,296 because 2 times 4 is 8, modulo 7 is 1. 331 00:25:22,156 --> 00:25:25,136 So 2 and 4 are called associates 332 00:25:25,136 --> 00:25:29,196 and they're basically 2 is the inverse of 4, 4 is the inverse of 2. 333 00:25:29,776 --> 00:25:32,776 Just like in the reals, 5 is the inverse of 1 fifth, 334 00:25:33,136 --> 00:25:34,616 1 fifth is the inverse of 5. 335 00:25:36,056 --> 00:25:41,716 But in this number system, the finite field modulo 7, 336 00:25:42,196 --> 00:25:44,876 2 is the inverse of 4, 4 is the inverse of 2. 337 00:25:44,876 --> 00:25:48,116 If we hit the number 3 338 00:25:48,116 --> 00:25:49,516 Right 339 00:25:49,516 --> 00:25:53,016 And I think the inverse of 3 is 5 340 00:25:53,016 --> 00:25:55,496 Because 3 times 5 is 15 341 00:25:55,496 --> 00:25:57,396 Which when divided by 7 342 00:25:57,396 --> 00:25:59,856 Has a remainder of 1 343 00:25:59,856 --> 00:26:02,396 So 2 and 4 are associates 344 00:26:02,396 --> 00:26:03,636 Inverses of each other 345 00:26:03,636 --> 00:26:04,936 3 and 5 are associates 346 00:26:04,936 --> 00:26:07,036 So what have we left? 347 00:26:07,356 --> 00:26:08,316 4 and 348 00:26:08,316 --> 00:26:10,076 6 349 00:26:10,076 --> 00:26:10,576 Sorry 350 00:26:10,576 --> 00:26:12,436 We have 351 00:26:12,436 --> 00:26:20,916 we have two and we said two and four three and five yes so we're three five and so six is kind of 352 00:26:20,916 --> 00:26:26,256 six is special you know why because you start going through well that's the only number left 353 00:26:26,256 --> 00:26:35,416 what the heck right well six before i do before i do the big reveal six modulo seven is six but 354 00:26:35,416 --> 00:26:42,036 it's also negative one as as it happens and this is a very interesting property because the number 355 00:26:42,036 --> 00:26:51,496 less than 1, it's its own inverse. Because negative 1 times negative 1 is 1. 6 times 6 is 36, 356 00:26:52,056 --> 00:27:01,576 which, you know, modulo 7, 7 times 5 is 35. So 36 minus 35, remainder 1. So 6 is its own inverse, 357 00:27:01,576 --> 00:27:07,676 and the prime, whatever the last number, like the largest number in your finite field, 358 00:27:07,676 --> 00:27:15,096 is always negative one and it's always its own inverse kind of cool yeah right it's like the 359 00:27:15,096 --> 00:27:19,596 and then that's the last that's the last one so we identified so it's it closes the field 360 00:27:19,596 --> 00:27:25,176 everyone has a buddy that's right everyone has a buddy and one obviously has a buddy we forgot 361 00:27:25,176 --> 00:27:31,856 we forgot we never want to forget one that's true it's very easy to do so everybody has a buddy 362 00:27:31,856 --> 00:27:35,016 and your buddy is your thread out of the cave 363 00:27:35,016 --> 00:27:35,756 right 364 00:27:35,756 --> 00:27:38,496 and it's like the way 365 00:27:38,496 --> 00:27:40,676 so that when you go to the bitcoin store you don't have to 366 00:27:40,676 --> 00:27:42,856 tell people hey can you make sure you prove 367 00:27:42,856 --> 00:27:44,956 to me that my guy has 368 00:27:44,956 --> 00:27:46,076 an inverse so that 369 00:27:46,076 --> 00:27:48,216 my point is back on the curve 370 00:27:48,216 --> 00:27:50,776 you have to trust that 371 00:27:50,776 --> 00:27:53,016 no matter what operations you do 372 00:27:53,016 --> 00:27:54,896 you will always be identifiable 373 00:27:54,896 --> 00:27:55,576 on that curve 374 00:27:55,576 --> 00:27:57,336 it's cool 375 00:27:57,336 --> 00:28:00,556 these inverses 376 00:28:00,556 --> 00:28:06,236 Like you're saying, when you're going to the Bitcoin store, it's basically when you're verifying a signature, you're doing this. 377 00:28:06,736 --> 00:28:07,536 This is what you're doing. 378 00:28:09,056 --> 00:28:12,116 Yes, you're verifying that the point exists. 379 00:28:12,116 --> 00:28:16,456 And when you're going to the Bitcoin store saying, hey, I have one Bitcoin, I want to spend it somewhere. 380 00:28:16,656 --> 00:28:22,296 The whole network is doing this math of finding the inverse to prove that you own the private key. 381 00:28:23,276 --> 00:28:23,616 That's right. 382 00:28:25,356 --> 00:28:26,156 Pretty sick. 383 00:28:26,716 --> 00:28:28,656 Now, are they really doing the inverse, though? 384 00:28:30,556 --> 00:28:32,056 That's the question, right? 385 00:28:32,096 --> 00:28:34,256 Because it's like they're not right. 386 00:28:34,516 --> 00:28:35,836 They're not really doing the inverse. 387 00:28:38,676 --> 00:28:39,196 Correct. 388 00:28:39,376 --> 00:28:41,296 Does that take us to the next point? 389 00:28:41,616 --> 00:28:43,936 Maybe we'll just put a pin in that one, right? 390 00:28:44,996 --> 00:28:48,876 Because a couple of primitives I think are really like, 391 00:28:48,876 --> 00:28:52,316 so I think that I do want to talk about the Euclidean algorithm. 392 00:28:52,456 --> 00:28:53,616 That's what I was trying to tee up next. 393 00:28:53,836 --> 00:28:54,576 Yeah, let's go there. 394 00:28:54,576 --> 00:28:56,276 So like none of this 395 00:28:56,276 --> 00:28:58,056 Euclidean algorithm is 396 00:28:58,056 --> 00:29:00,576 A fancy name for something 397 00:29:00,576 --> 00:29:02,736 We all started doing when we were 398 00:29:02,736 --> 00:29:05,036 Seven years old which is long division 399 00:29:05,036 --> 00:29:08,636 Right or I'd say when we did 400 00:29:08,636 --> 00:29:10,076 Long division when we were seven 401 00:29:10,076 --> 00:29:11,656 We were kind of 402 00:29:11,656 --> 00:29:14,496 We were doing the Euclidean algorithm without 403 00:29:14,496 --> 00:29:15,596 Without the name 404 00:29:15,596 --> 00:29:17,196 Would you say? 405 00:29:18,476 --> 00:29:20,616 Yeah you bring it to a level 406 00:29:20,616 --> 00:29:22,496 When you get to the point of long division 407 00:29:22,496 --> 00:29:23,596 Like third fourth grade 408 00:29:23,596 --> 00:29:26,176 you've done your addition 409 00:29:26,176 --> 00:29:27,596 you've done multiplication 410 00:29:27,596 --> 00:29:29,216 you started doing some multiplication 411 00:29:29,216 --> 00:29:32,056 and now what you're doing with long division is you're combining addition 412 00:29:32,056 --> 00:29:33,116 and multiplication 413 00:29:33,116 --> 00:29:35,436 to kind of undo the whole thing 414 00:29:35,436 --> 00:29:37,556 yes and we did this in a field 415 00:29:37,556 --> 00:29:39,136 we didn't know it was a field 416 00:29:39,136 --> 00:29:41,856 we didn't know that all this stuff was nice 417 00:29:41,856 --> 00:29:43,776 but we did it in a field called real numbers 418 00:29:43,776 --> 00:29:45,596 right 419 00:29:45,596 --> 00:29:46,536 and 420 00:29:46,536 --> 00:29:49,576 with well using 421 00:29:49,576 --> 00:29:50,276 integers 422 00:29:50,276 --> 00:29:52,176 really just using 423 00:29:52,176 --> 00:29:57,956 we using integers as coefficients maybe we could say we did it in a ring a ring of integers but 424 00:29:57,956 --> 00:30:01,776 we didn't know about the structure we didn't know why this would all work 425 00:30:01,776 --> 00:30:10,236 frankly not to get to a field again but we don't we don't realize that a number is actually a 426 00:30:10,236 --> 00:30:16,256 linear combination it's a polynomial a number is a polynomial which in polynomials form rings and 427 00:30:16,256 --> 00:30:31,136 And this is actually why a lot of this works as well, meaning like the number of 5,262 is 2 plus 6 times 10 plus – I already forgot the number, but you get it. 428 00:30:32,036 --> 00:30:33,956 It's a linear combination of powers of 10. 429 00:30:34,916 --> 00:30:40,296 And we have powers of 2 in our binary numbers that are just – these are polynomials. 430 00:30:40,496 --> 00:30:41,156 They are a ring. 431 00:30:41,156 --> 00:30:47,916 we do the Euclidean algorithm on a on a we're pulling apart that polynomial with the base 432 00:30:47,916 --> 00:30:52,896 would it be fair to say Euclidean algorithm like uh it's basically 433 00:30:52,896 --> 00:30:57,276 maybe another term from like fourth fifth grade math like greatest common denominator 434 00:30:57,276 --> 00:31:04,096 it's a way of finding that it's a way of finding that and usually when we were in third fourth 435 00:31:04,096 --> 00:31:10,716 grade it was heartbreaking to have to find a remainder we're like oh no no I hate why isn't 436 00:31:10,716 --> 00:31:17,836 it nice and easy even and it all goes in nicely yeah right but it's basically finding whether or 437 00:31:17,836 --> 00:31:25,516 finding a greatest and in in our parlance we call it greatest common divisor so when i mentioned 438 00:31:25,516 --> 00:31:31,916 relatively prime before basically when two numbers are relatively prime it means there's no more 439 00:31:31,916 --> 00:31:35,536 there's no more division you can do with any of them you're sort of at the bottom of the barrel 440 00:31:35,536 --> 00:31:40,656 of how much you can reduce your system but when there are greatest common divisors this can 441 00:31:40,656 --> 00:31:55,376 This comes up a lot like something simple like in the Pythagorean theorem, right, of trying to sort of solve a lot of that stuff that, you know, you have a 3, 4, 5 triangle, 3 squared plus 4 squared equals 5 squared. 442 00:31:56,056 --> 00:31:57,656 But like that scales. 443 00:31:57,916 --> 00:32:01,896 Like, you know, that can all scale with size. 444 00:32:02,016 --> 00:32:05,316 So 3, 4, 5 and a 6, 8, 10 are kind of the same thing. 445 00:32:05,316 --> 00:32:14,116 And so we only want to really, in our systems when we do this, we only want to operate in the system where there are no common divisors, right? 446 00:32:14,176 --> 00:32:16,256 That's just sort of like a, it's not an axiom. 447 00:32:16,356 --> 00:32:26,256 It's more of just like it makes it faster and easier to just always be, to not be that redundant, right? 448 00:32:26,416 --> 00:32:31,616 To say, well, it's kind of similar to modular arithmetic. 449 00:32:31,616 --> 00:32:41,436 It's just we're operating in the lowest kind of lowest atomic element of being able to divide these numbers. 450 00:32:41,436 --> 00:32:50,096 So what the Euclidean algorithm does is it uses this recursive – that's what the algorithm kind of means. 451 00:32:50,176 --> 00:33:02,016 It uses this recursive division of finding remainders to ultimately get sort of to the end and find out if there's a greatest common divisor between two numbers. 452 00:33:02,696 --> 00:33:04,376 Well, find out what it is. 453 00:33:05,036 --> 00:33:09,856 And if it's one, that means those two numbers are relatively prime. 454 00:33:09,856 --> 00:33:12,616 So that's all to say 455 00:33:12,616 --> 00:33:13,916 Backtrack a little bit 456 00:33:13,916 --> 00:33:16,476 I get a couple numbers like 6 and 7 457 00:33:16,476 --> 00:33:18,116 I know they're relatively prime 458 00:33:18,116 --> 00:33:20,216 I know that if I divide 7 into 6 459 00:33:20,216 --> 00:33:21,336 I get a remainder of 1 460 00:33:21,336 --> 00:33:22,436 Boom 461 00:33:22,436 --> 00:33:24,316 That's how I 462 00:33:24,316 --> 00:33:25,896 Their greatest common divisor is 463 00:33:25,896 --> 00:33:27,336 I know they have no common divisors 464 00:33:27,336 --> 00:33:31,016 But if I'm dealing with numbers that are hundreds of digits long 465 00:33:31,016 --> 00:33:34,056 It would be really nice 466 00:33:34,056 --> 00:33:36,196 If there was a way to find out 467 00:33:36,196 --> 00:33:36,876 If there were greatest 468 00:33:36,876 --> 00:33:40,376 if there was common divisibility, 469 00:33:41,396 --> 00:33:43,356 numbers of common divisibility among them. 470 00:33:43,536 --> 00:33:45,156 I'm happy you started with six and seven 471 00:33:45,156 --> 00:33:48,076 for the 12-year-olds that are in the audience. 472 00:33:48,816 --> 00:33:49,716 You did it. 473 00:33:50,156 --> 00:33:51,676 Oh, they should stop the podcast right now. 474 00:33:51,696 --> 00:33:52,596 Yeah, it's all over. 475 00:33:52,596 --> 00:33:55,276 No, but this is a really important point though, right? 476 00:33:55,376 --> 00:33:58,416 Is that being able to quickly be able to derive 477 00:33:58,416 --> 00:33:59,776 and understand these properties 478 00:33:59,776 --> 00:34:02,976 is just very important for doing this at scale. 479 00:34:02,976 --> 00:34:10,676 and so while the greatest so the euclidean algorithm is a powerful way that this is 480 00:34:10,676 --> 00:34:14,716 this is a rich thing in and of itself because um 481 00:34:14,716 --> 00:34:21,316 a it's a powerful way of finding common divisors which is actually really important 482 00:34:21,316 --> 00:34:27,556 with that with a common in order to know that your field is real is fine you have like a finite 483 00:34:27,556 --> 00:34:32,756 field, you have to know that any two numbers are relatively prime. That gives you the property of 484 00:34:32,756 --> 00:34:37,176 the, that's part of what gives you the property of the inverse. So like the, you already know 485 00:34:37,176 --> 00:34:41,076 there's a fly in the ointment. If there's a greatest common divisor of any two elements in 486 00:34:41,076 --> 00:34:45,776 your finite field that are not one. So that's like an important thing to be, that's one way to verify, 487 00:34:46,416 --> 00:34:52,296 it's one way to verify this, right? It's also that you can, um, 488 00:34:52,296 --> 00:35:02,356 there's this um i was telling you before we started like when i was learning all this a couple 489 00:35:02,356 --> 00:35:08,296 like a year ago or two and i was like really looking to try to teach it and grok it you know 490 00:35:08,296 --> 00:35:13,396 one of the things i like i like in a textbook is if there's like a code implementation that really 491 00:35:13,396 --> 00:35:17,536 shows me and this is like part of how we got here to begin with because like bitcoin's code 492 00:35:17,536 --> 00:35:24,776 implementation is beautiful it's like so informative libsec p not so much right and um 493 00:35:24,776 --> 00:35:30,896 it's like what the hell right so if you ever ask like i asked chat gpt that was like the best ai 494 00:35:30,896 --> 00:35:38,996 at the time i said you know make me a function for the euclidean algorithm um i encourage you 495 00:35:38,996 --> 00:35:43,156 guys to do that because and it's what's so interesting about it is it'll give you literally 496 00:35:43,156 --> 00:35:51,756 one of the shortest, it'll give you a one line formula. This is what I got one line that is like 497 00:35:51,756 --> 00:36:00,496 actually not maybe 30 characters long. And I didn't find that very helpful. Right. I was like, 498 00:36:00,556 --> 00:36:06,416 what, what, what is going on? And then it's like, I wonder if you need to go to school for computer 499 00:36:06,416 --> 00:36:13,296 science to understand this right but the more that's ironic that the more i repeated the more 500 00:36:13,296 --> 00:36:19,236 you repeat going through this and the more i did pen and pencil exercises on the euclidean algorithm 501 00:36:19,236 --> 00:36:24,216 just like any two numbers you just do your pen and pencil exercises and i don't know how many it 502 00:36:24,216 --> 00:36:33,016 takes but at some point it clicks that it's this recursiveness is uh it's that it's because it's 503 00:36:33,016 --> 00:36:37,856 recursive is very clever that you could do it all in one line and just say, just, you know, 504 00:36:38,256 --> 00:36:46,276 go until essentially go until, you know, until you can't go anymore. But, um, and then output 505 00:36:46,276 --> 00:36:55,816 either one or whatever the final number is. Right. And if like, it's a lot of what we do here 506 00:36:55,816 --> 00:37:01,536 is about going in, it's inside out. The way you learn is inside out, not outside in, 507 00:37:01,536 --> 00:37:06,516 You go to the bottom of the iceberg and you start climbing your way out. 508 00:37:07,196 --> 00:37:19,476 And so I would encourage people to do exercises on the Euclidean algorithm and inside out with real numbers. 509 00:37:19,476 --> 00:37:26,276 Because this is, again, it's the way, it's the beginning of understanding why, A, trusting. 510 00:37:26,416 --> 00:37:27,416 I mean, trust is a bad word. 511 00:37:27,416 --> 00:37:33,656 It's the best word I have right now for saying, how do you know it actually works on numbers you can't do this on? 512 00:37:33,936 --> 00:37:36,156 You can't do it by hand or on a computer. 513 00:37:37,676 --> 00:37:45,816 So I don't know if you had any more on that one. 514 00:37:52,796 --> 00:37:54,256 Rob is deeply thinking. 515 00:37:54,256 --> 00:38:04,356 this is deeply thinking and a microphone on mute when i'm talking uh so looking through this like 516 00:38:04,356 --> 00:38:08,876 i think the pen and paper exercise is good if people want to kind of follow along and i think 517 00:38:08,876 --> 00:38:15,756 the study guide is like a great way to like tee up the context around it um do we want to tie it 518 00:38:15,756 --> 00:38:23,876 back to finding the inverse with the extended euclidean algorithm yes yeah we should do that 519 00:38:23,876 --> 00:38:27,676 I call it 520 00:38:27,676 --> 00:38:29,416 There's a lot of things you can call 521 00:38:29,416 --> 00:38:31,396 There's a thing called Bayes-Out's identity 522 00:38:31,396 --> 00:38:33,436 Which 523 00:38:33,436 --> 00:38:34,976 So basically when you do the 524 00:38:34,976 --> 00:38:36,316 It's like when you do a long division 525 00:38:36,316 --> 00:38:38,896 You can then put that process in reverse 526 00:38:38,896 --> 00:38:40,056 To find 527 00:38:40,056 --> 00:38:41,676 The actual 528 00:38:41,676 --> 00:38:45,156 You can come back to the two numbers 529 00:38:45,156 --> 00:38:47,436 You were trying to 530 00:38:47,436 --> 00:38:49,216 Reduce to begin with 531 00:38:49,216 --> 00:38:50,796 Right 532 00:38:50,796 --> 00:38:52,936 How would you approach this 533 00:38:52,936 --> 00:39:00,876 explaining this in words well this is just the idea that um you since the nature of multiplication 534 00:39:00,876 --> 00:39:05,936 like we're talking greatest common divisor like divisor denominator whatever like you're taking 535 00:39:05,936 --> 00:39:10,876 a system you it can go both ways so you can start with the numbers that you're looking to find the 536 00:39:10,876 --> 00:39:15,796 greatest common divisor or you can start with what that number is and work backwards to find the 537 00:39:15,796 --> 00:39:21,996 original numbers again you're basically taking both sides uh you can go a to b or b to a and 538 00:39:21,996 --> 00:39:23,576 Those are the same idea. 539 00:39:23,836 --> 00:39:37,556 But I guess the powerful thing you have with Bayes-Out's identity is that what you have – if you do have two numbers that resolve sort of – that are relatively prime, that the algorithm tells you are relatively prime, right? 540 00:39:37,556 --> 00:39:42,996 Then what you now have is a linear combination of those two original numbers that resolve to the identity. 541 00:39:42,996 --> 00:39:48,756 and so that you can sort of see the, you know, 542 00:39:49,496 --> 00:39:54,696 it gives you now powerful algebraic ability to eliminate, 543 00:39:55,296 --> 00:39:58,836 you know, to eliminate like a lot of factors 544 00:39:58,836 --> 00:40:02,116 when you're now doing math with those numbers, 545 00:40:02,116 --> 00:40:04,096 which is really what you're doing to begin with. 546 00:40:05,536 --> 00:40:06,956 I know that probably didn't connect, 547 00:40:06,956 --> 00:40:11,456 but it's just like what Bayes-Out's identity says 548 00:40:11,456 --> 00:40:16,836 is that two numbers that are relatively prime, 549 00:40:17,016 --> 00:40:19,896 you can have any linear combination of those two numbers 550 00:40:19,896 --> 00:40:23,916 and it'll equal one modulo that prime. 551 00:40:24,256 --> 00:40:25,796 So in the modulo arithmetic world, 552 00:40:26,216 --> 00:40:30,896 that's, you know, so anytime you have numbers, 553 00:40:31,216 --> 00:40:32,496 anytime you have anything, 554 00:40:32,796 --> 00:40:35,336 when you operate it and it resolves to one, 555 00:40:35,536 --> 00:40:36,276 i.e. your identity, 556 00:40:36,476 --> 00:40:37,656 you're going to be able to use that 557 00:40:37,656 --> 00:40:40,476 to essentially make the math easier down the line. 558 00:40:41,456 --> 00:40:49,076 um i don't know if that really covers i don't know if that really covers it i'm i'm pulling 559 00:40:49,076 --> 00:40:53,536 up right now to do a but that leads to format's little theorem which i think is like one of the 560 00:40:53,536 --> 00:41:01,036 most important things anybody yeah yeah and i think like no in the bigger picture we're talking 561 00:41:01,036 --> 00:41:07,916 about this large finite field for all the bitcoin stuff but like let's just talk about it at the 562 00:41:07,916 --> 00:41:16,316 simplest level if you were to take very small numbers right so let's let's start with um 563 00:41:16,316 --> 00:41:20,176 i have here i just please don't choose six and seven please don't choose six and seven 564 00:41:20,176 --> 00:41:27,396 let's do uh the inverse of three modulo 17 so what you're trying to do is you're trying to say 565 00:41:27,396 --> 00:41:35,336 three times what equals one mod 17 right so you have this number 17 and what you do is you could 566 00:41:35,336 --> 00:41:42,896 take well that's three times five plus two it's because three times five is 15 plus two is 17 567 00:41:42,896 --> 00:41:48,636 right yeah 17 is divided by three uh 17 divided by three is five with the remainder of two 568 00:41:48,636 --> 00:41:54,916 and so you have three now equals two times one plus one and then you hit a remainder of one 569 00:41:54,916 --> 00:42:00,576 you stop you found the answer now phase two is to build it back up you're kind of walking backwards 570 00:42:00,576 --> 00:42:07,656 wait hold on hold on hold on yeah hold on um your remainder of two basically means that it's not 571 00:42:07,656 --> 00:42:11,536 your inverse though you what will you really keep on going you have to keep on going until 572 00:42:11,536 --> 00:42:16,416 the recursive nature yes that's right so you start with 17 right and you're trying to say 573 00:42:16,416 --> 00:42:22,776 again the question is is so you do a euclidean algorithm between you do you try use the euclidean 574 00:42:22,776 --> 00:42:28,616 algorithm to try to find the greatest common divisor of three and 17 and this is like the tool 575 00:42:28,616 --> 00:42:34,256 this is very hard to talk through right but this is the tool of help helping you find the inverse 576 00:42:34,256 --> 00:42:41,416 of three with respect to 17 in that finite field bingo and so just to walk through it again the 577 00:42:41,416 --> 00:42:49,536 question is if you're taking the inverse of well so you're saying in the problem is what number 578 00:42:49,536 --> 00:42:57,016 do i multiply by three in modulo 17 do i get one and so you say okay well i have this 17 number 579 00:42:57,016 --> 00:43:01,916 equals. And what you do immediately is you start factoring out, you get three times five. 580 00:43:02,416 --> 00:43:07,076 Well, I get, well, no, because I get a remainder of two. I get a remainder of negative two actually 581 00:43:07,076 --> 00:43:11,916 with three times five, which is five. Well, it's the first step. So I get a remainder of 15, 582 00:43:11,916 --> 00:43:17,456 I get a remainder of 15, right? Negative two is negative two is 15 in this system. Right. So I 583 00:43:17,456 --> 00:43:21,336 get, yes, there you go. Right. So you have to keep on going. Right. So that's the first step. 584 00:43:21,396 --> 00:43:25,616 So if you're doing this with pen and paper, you have to keep on going. Right. And, um, 585 00:43:25,616 --> 00:43:32,376 You can then go one more lap and say, well, 3 divided by 2 is 1 with a remainder of 1. 586 00:43:32,676 --> 00:43:34,476 And you hit a remainder of 1, so then you stop. 587 00:43:35,776 --> 00:43:37,156 So what does that tell you, though? 588 00:43:38,836 --> 00:43:40,216 Let's just zoom out for a second. 589 00:43:40,376 --> 00:43:43,156 In the modulo 17, I just know the answer is 6. 590 00:43:43,336 --> 00:43:47,416 I know 3 times 6 is 18, which gives me a remainder of 1. 591 00:43:47,576 --> 00:43:53,116 So the real question is how does the – I don't know if this can be talked through or if it's possible without a pen and paper. 592 00:43:53,116 --> 00:43:59,056 But like how does the Euclidean algorithm in general get you to that inverse? 593 00:44:01,036 --> 00:44:22,056 And it's – I think what it comes down to is when you're doing it, you realize that one of the things you're – the base thing you're multiplying by is essentially going to be – you can zero it out because it has – you can zero it out because it has – it's a multiple of your modulus. 594 00:44:23,116 --> 00:44:23,376 Right. 595 00:44:23,656 --> 00:44:23,896 Right. 596 00:44:24,216 --> 00:44:32,956 It's like, and then what's left, when you finally get to a number that's lower than 17, that's going to be your inverse. 597 00:44:32,956 --> 00:44:37,496 And maybe I do need to code up an example here. 598 00:44:37,496 --> 00:44:40,696 Three times six equals 17 plus one. 599 00:44:41,196 --> 00:44:42,236 There's your remainder one. 600 00:44:42,596 --> 00:44:43,036 That's right. 601 00:44:43,116 --> 00:44:44,496 So I know the answer is six. 602 00:44:44,876 --> 00:44:49,216 It's very hard to take people through the steps of this thing to see that. 603 00:44:49,216 --> 00:44:53,356 we're gonna have to upgrade the production value and get a nice little interactive whiteboard going 604 00:44:53,356 --> 00:44:58,516 here yeah i'm on i'm like i'm i'm on it i'm thinking about it that's why like the youtube 605 00:44:58,516 --> 00:45:06,876 series is was designed to support this so you can see examples and you know when i it's like when we 606 00:45:06,876 --> 00:45:12,216 sort of step in shit like this that's when i discover what what's needed yeah the way i would 607 00:45:12,216 --> 00:45:16,996 view it is like if you're viewing like this this field in this system is like a machine what you're 608 00:45:16,996 --> 00:45:22,236 doing is you're labeling each part and disassembling it and reassembling it like ikea furniture like 609 00:45:22,236 --> 00:45:26,476 like that you know that's what like it's ikea furniture a field is like ikea furniture every 610 00:45:26,476 --> 00:45:29,956 single piece has a piece that you click in and there's instructions on how they all interact 611 00:45:29,956 --> 00:45:34,936 with each other and what we're doing in these operations is we're provably showing that you 612 00:45:34,936 --> 00:45:43,876 can take it apart put it back together and kind of complete the system yes and i think i think i'm 613 00:45:43,876 --> 00:45:45,376 Debating whether to do this right now 614 00:45:45,376 --> 00:45:46,156 But I'm going to do it 615 00:45:46,156 --> 00:45:46,916 I want to read 616 00:45:46,916 --> 00:45:47,876 I want to read the 617 00:45:47,876 --> 00:45:49,476 What I call the Steiner's lens 618 00:45:49,476 --> 00:45:51,616 Blurb from the 619 00:45:51,616 --> 00:45:52,516 You know 620 00:45:52,516 --> 00:45:53,416 From the manual 621 00:45:53,416 --> 00:45:54,456 From the study guide 622 00:45:54,456 --> 00:45:55,836 Because I just think it explains 623 00:45:55,836 --> 00:45:57,576 It finally does put words 624 00:45:57,576 --> 00:45:58,416 And explains like 625 00:45:58,416 --> 00:45:59,756 What I think is important here 626 00:45:59,756 --> 00:46:01,356 So I'm going to do it 627 00:46:01,356 --> 00:46:01,816 Okay 628 00:46:01,816 --> 00:46:14,192 So basically Steiner Distinguishes between A finished thought Which is like a proposition We contemplate from the outside and a thinking activity which is a process we follow through transformations 629 00:46:14,652 --> 00:46:16,192 So the thinking activity is like the algorithm. 630 00:46:17,392 --> 00:46:19,892 So an algorithm is closer to the latter. 631 00:46:20,392 --> 00:46:23,712 It is a living thinking, not a static truth. 632 00:46:24,372 --> 00:46:29,612 When you follow the Euclidean algorithm, you're not contemplating a fixed fact. 633 00:46:30,272 --> 00:46:34,432 You're participating in a thinking process that unfolds in time. 634 00:46:35,192 --> 00:46:38,492 Each step transforms the problem into a simpler version of itself. 635 00:46:39,192 --> 00:46:51,832 So in the example here, the greatest common divisor of 252 and 105 can be reduced to the greatest common divisor of 105 and 42. 636 00:46:51,832 --> 00:46:54,792 I'll just pause for a second to say 637 00:46:54,792 --> 00:46:55,852 because you can 638 00:46:55,852 --> 00:47:00,772 you can get through that 639 00:47:00,772 --> 00:47:04,132 that should be a step that people can do self-evidently 640 00:47:04,132 --> 00:47:06,712 but by doing the division 641 00:47:06,712 --> 00:47:10,452 105 goes into 252 twice with a remainder of 42 642 00:47:10,452 --> 00:47:11,712 so they have this 643 00:47:11,712 --> 00:47:15,912 you have the same greatest common divisor between 252 and 105 644 00:47:15,912 --> 00:47:17,672 as 105 and 42 645 00:47:17,672 --> 00:47:21,712 and then you do that step again and you get a GCD of 42 646 00:47:21,712 --> 00:47:30,132 and 21, which maybe some people recognize as 21 because 21 times 2 is 42. So basically, again, 647 00:47:30,172 --> 00:47:34,832 you understand the algorithm not by looking at it from the outside, but by doing it, 648 00:47:35,312 --> 00:47:39,232 entering into the process and following it through each transformation until it reaches 649 00:47:39,232 --> 00:47:45,792 its conclusion. That's what I wanted to read from that. And what I want to kind of like also 650 00:47:45,792 --> 00:47:52,092 mention is that I just, I've always thought about this thing with Rudolf Steiner. My kids 651 00:47:52,092 --> 00:47:58,132 are in Waldorf schools. And one of the things I noticed very early on with my kids in that school, 652 00:47:58,132 --> 00:48:06,532 as it applies to this, is every kid in Waldorf school, an adult, if you ever go there, 653 00:48:06,852 --> 00:48:15,252 is knitting. Knitting is a big part of the education. And part of it, what they talk about 654 00:48:15,252 --> 00:48:16,972 is emphasizing working with your hands. 655 00:48:17,372 --> 00:48:21,192 But what I saw was, particularly with crochet, 656 00:48:21,992 --> 00:48:26,672 is that you trust a pattern and then you do it. 657 00:48:26,992 --> 00:48:31,152 And you don't know until you're done that it was going to work. 658 00:48:31,152 --> 00:48:36,552 But these people who look like hippies that sit around knitting all the time, 659 00:48:36,632 --> 00:48:39,932 they actually have high conviction in this algorithm. 660 00:48:40,752 --> 00:48:40,812 Right. 661 00:48:41,192 --> 00:48:44,652 And I remember seeing that for myself and telling people, 662 00:48:44,652 --> 00:48:53,552 you realize you guys are teaching math here without realizing it because this is like you're teaching people to trust an algorithm that's going to lead to this. 663 00:48:54,012 --> 00:48:56,312 And, you know, there's a reason why it always works. 664 00:48:58,412 --> 00:49:05,652 And, you know, if you if your pattern didn't didn't come out the way it was supposed to do, you messed a step up in that process. 665 00:49:06,532 --> 00:49:08,152 But you learn by doing it. 666 00:49:08,192 --> 00:49:11,052 You learn by putting yourself through it. 667 00:49:11,052 --> 00:49:16,492 Just because we have computers and AI doesn't mean you can shortcut that process of learning. 668 00:49:17,612 --> 00:49:21,752 And so I would think of it like knitting a sweater. 669 00:49:24,192 --> 00:49:27,952 All you know is the next step and the next step and the next step and the next step. 670 00:49:28,212 --> 00:49:30,052 You're just repeating the same thing. 671 00:49:31,192 --> 00:49:37,432 But you do understand the structure of the thing and why it's going to – you do sort of understand why it does turn out. 672 00:49:37,432 --> 00:49:47,212 Because why would you spend your time playing around with needles and threads like that if you didn't know it was going to turn out cool in the way you thought it was? 673 00:49:48,652 --> 00:49:53,312 And so it's like it's internally like you it's it's like internally to you as a human being. 674 00:49:53,812 --> 00:50:02,172 You get to gain the experience of, you know, this algorithm, which traces back to YouTube. 675 00:50:02,172 --> 00:50:06,112 Like no one's improved on this in 2000 plus years. 676 00:50:06,652 --> 00:50:07,132 Right. 677 00:50:07,432 --> 00:50:15,452 it's pretty remarkable and at the base of it again it's just really trying at the base of it 678 00:50:15,452 --> 00:50:20,012 we're trying to find out if two numbers have a share a common divisor yeah but it gets you but 679 00:50:20,012 --> 00:50:25,772 it also helps it's so powerful that it also you know this traces back to the inverse problem and 680 00:50:25,772 --> 00:50:29,992 i often think too that like sometimes like the reason why something like this like is thousands 681 00:50:29,992 --> 00:50:33,472 of years ago and we haven't really been able to iteratively prove on it is because it's just 682 00:50:33,472 --> 00:50:39,112 it's so simple and back then you did not have the aid of technology to try and like accelerate and 683 00:50:39,112 --> 00:50:44,052 complicate things and get more abstract you had to just sit there and think about it that was 684 00:50:44,052 --> 00:50:50,312 really your yep that was really what you had and it's a beautiful property um it's how i got to 685 00:50:50,312 --> 00:50:55,492 grow up too like in life i feel like kids without the internet got to grow up really sitting and 686 00:50:55,492 --> 00:51:02,012 thinking about the things they're doing like that and all the more reason why i think it's good to 687 00:51:02,012 --> 00:51:11,672 emphasize um spending this kind of time if you know again you it's this is all how the substitute 688 00:51:11,672 --> 00:51:18,092 for being able to go to the bitcoin store and ask to prove to you that your key exists yeah and should 689 00:51:18,092 --> 00:51:23,692 yeah should we take this now too because we keep there's a reason why we mentioned this a little 690 00:51:23,692 --> 00:51:33,512 bit earlier we're doing modulo 7 modulo 17 like we're very bold you were very bold and going for 691 00:51:33,512 --> 00:51:40,152 17 yeah well like the idea though the idea of why we're picking these numbers and we mentioned it 692 00:51:40,152 --> 00:51:46,152 briefly earlier is that these are the what's unique about these numbers are they are prime numbers 693 00:51:46,152 --> 00:51:50,832 right do we want to take this maybe a little bit to fermat's little theorem 694 00:51:50,832 --> 00:51:59,292 yes we're gonna go to fermat's little theorem which um everyone knows about fermat's last 695 00:51:59,292 --> 00:52:06,172 theorem but frankly fermat's last theorem isn't that useful to us fermat's little theorem oh my 696 00:52:06,172 --> 00:52:14,932 god one of the most powerful incredible um little factoids that this is what we we're now 697 00:52:14,932 --> 00:52:19,832 possibly we're taking the pin out of are you really calculating the inverse it's like what 698 00:52:19,832 --> 00:52:24,192 are you actually doing? Like that's, we're kind of uncorking that now to revisit that point from 699 00:52:24,192 --> 00:52:29,672 earlier. So you remember when we said, boy, anything that resolves to the identity is very 700 00:52:29,672 --> 00:52:35,172 powerful because we can then eliminate a lot of the math once we, cause you know, you multiply by 701 00:52:35,172 --> 00:52:43,692 one over and over again, guess what? Nothing changes, right? So, um, it's nice to just multiply 702 00:52:43,692 --> 00:52:51,452 things by one i wish i could do it all the time but um you know take a number like um so but let's 703 00:52:51,452 --> 00:52:58,932 take this field modulo seven right and let's just take an element of our field that we know is a 704 00:52:58,932 --> 00:53:04,672 generator i think let's just do you want to just prove that two do we know if two is a generator 705 00:53:04,672 --> 00:53:09,412 should we just go through that let's just go through the exercise go through the here so we 706 00:53:09,412 --> 00:53:19,312 get two and two times two is four right two times and then so we get two four and six well i'm this 707 00:53:19,312 --> 00:53:24,672 is on the additive side on the multiplicative side we have two four eight which is one right 708 00:53:24,672 --> 00:53:31,072 so two is not a generator because it only gives you two four and one it stops there so let's we 709 00:53:31,072 --> 00:53:35,712 got and you have a closed loop and it's not all of the points so that's not correct and for the 710 00:53:35,712 --> 00:53:39,972 benefit of the audience once i get to once you get to one you're just starting the cycle again 711 00:53:39,972 --> 00:53:45,672 so two one times two again is two two times two again is four four times two again is you're kind 712 00:53:45,672 --> 00:53:51,232 of on rails you really can't get outside that that cycle yes so okay two is not a generator 713 00:53:51,232 --> 00:53:57,912 and there's a this is this is in a number theory concept called the primitive root primitive root 714 00:53:57,912 --> 00:54:03,592 it's not that easy it's not easy to find so that's why we define the generator in bitcoin by the way 715 00:54:03,592 --> 00:54:06,512 when you go to the GitHub, you see the big capital G 716 00:54:06,512 --> 00:54:07,432 is defined already. 717 00:54:08,312 --> 00:54:10,852 It's the smallest number that can be the generator. 718 00:54:11,392 --> 00:54:12,672 Let's now try 3. 719 00:54:13,392 --> 00:54:14,552 I have a good feeling about it. 720 00:54:15,172 --> 00:54:15,512 Me too. 721 00:54:15,812 --> 00:54:18,192 3 times 3 is 9, which is 2. 722 00:54:18,352 --> 00:54:21,232 3 goes to 2, which goes to 6. 723 00:54:21,532 --> 00:54:22,312 2 times 3 is 6. 724 00:54:22,312 --> 00:54:23,712 We get 3 to 2 to 6. 725 00:54:24,732 --> 00:54:26,832 Then we go to 18, 726 00:54:27,032 --> 00:54:27,452 which is 727 00:54:27,452 --> 00:54:29,972 in modulo 7, 728 00:54:30,152 --> 00:54:30,752 which is 4. 729 00:54:30,992 --> 00:54:32,452 3 to 2 to 6 is 4. 730 00:54:32,452 --> 00:54:40,192 four times oh no yeah so then we go to five yeah four goes to 12 right did we do this right 731 00:54:40,192 --> 00:54:45,752 are you sure we did this right three to six three times we're doing generator three yes 732 00:54:45,752 --> 00:54:53,012 three to six yeah so we go three six four to five wait sorry so three times three is nine 733 00:54:53,012 --> 00:54:59,752 three goes to seven is two three to two times three is six modulo seven is just six yes so i 734 00:54:59,752 --> 00:55:02,252 I just want to keep saying it. 735 00:55:02,292 --> 00:55:07,812 We go 3 to 2 to 6, right? 736 00:55:08,032 --> 00:55:15,412 And you should be in your mind's eye, have the numbers 1, 2, 3, 4, 5, and 6 and crossing them off as we identify this. 737 00:55:15,432 --> 00:55:16,232 So we started with 3. 738 00:55:16,732 --> 00:55:19,252 We went to 2 and then to 6. 739 00:55:19,312 --> 00:55:20,152 So those are crossed off. 740 00:55:20,152 --> 00:55:22,172 And then 6 times 3 is 18. 741 00:55:22,672 --> 00:55:23,792 Which is 4. 742 00:55:24,432 --> 00:55:26,392 Modulo 7 is 4. 743 00:55:26,692 --> 00:55:28,792 Yeah, so it's 3 to 2 to 6 to 4. 744 00:55:28,792 --> 00:55:30,872 4 times 3 is 12 745 00:55:30,872 --> 00:55:33,052 that's modulo 6 is 5 746 00:55:33,052 --> 00:55:35,752 so 3 to 2 to 6 to 4 to 5 747 00:55:35,752 --> 00:55:38,212 and then 5 times 3 is 15 748 00:55:38,212 --> 00:55:39,152 modulo 7 749 00:55:39,152 --> 00:55:40,172 is 1 750 00:55:40,172 --> 00:55:42,192 so boom 751 00:55:42,192 --> 00:55:44,092 so 3 is the generator 752 00:55:44,092 --> 00:55:46,712 3 definitely gives us all 6 elements 753 00:55:46,712 --> 00:55:48,592 you can create all 6 elements 754 00:55:48,592 --> 00:55:51,412 and remember this is the same thing we talk about 755 00:55:51,412 --> 00:55:52,612 when we say that 756 00:55:52,612 --> 00:55:53,812 the 757 00:55:53,812 --> 00:55:55,652 your 758 00:55:55,652 --> 00:56:00,952 points in the elliptic curve can all be generated by the big generator point, which means when you 759 00:56:00,952 --> 00:56:06,452 have a private key, any other number in the field multiplies to a point. That's all we just did there 760 00:56:06,452 --> 00:56:12,532 with prime being seven, right? Okay. So back to the... So if you have a... 761 00:56:14,252 --> 00:56:17,352 I don't know if it was necessary for us to just do that, but... 762 00:56:17,352 --> 00:56:18,072 It's a good little refresh. 763 00:56:18,072 --> 00:56:18,512 It was fun. 764 00:56:18,752 --> 00:56:23,632 Because we talked about generators last episode. It's a good way just to kind of like nudge the 765 00:56:23,632 --> 00:56:29,632 concept real quick and we can go on to fermat so okay so in fermat we basically say any number 766 00:56:29,632 --> 00:56:39,052 any number in that field to the power of p minus one is one and so that's really that's really 767 00:56:39,052 --> 00:56:43,592 interesting you can go through the math on that but like we can just basically say okay we can 768 00:56:43,592 --> 00:56:50,532 actually say two to the sixth power is one when we we really did show it because we went to remember 769 00:56:50,532 --> 00:56:59,432 we went 2, 4, 8. We just do it twice. 2, 4, 8, 2, 4, 8. That's 1, right? 2, 4, 1, 2, 4, 1. So 2 to the 6th 770 00:56:59,432 --> 00:57:07,592 power is 1. We showed the 3 being a generator. Clearly, 3 to the 6th power is 1, right? And then 771 00:57:07,592 --> 00:57:14,912 it gets kind of easy because if 2 to the 6th is 1, then 4 to the 6th is 1 also. Why? Well, because 772 00:57:14,912 --> 00:57:21,652 four is two squared and so bingo you know factor it you can just do the math and you're multiplying 773 00:57:21,652 --> 00:57:28,292 one by itself now a bunch this is an important point is that when you find that point you can 774 00:57:28,292 --> 00:57:33,592 take that base number and multiply it again but you're just if you factor it out you're just doing 775 00:57:33,592 --> 00:57:38,672 one times one times one however many times you're putting it together and this gets to the power 776 00:57:38,672 --> 00:57:40,512 Because if 2 to the 6th is 1 777 00:57:40,512 --> 00:57:42,892 Then 2 to the 600th is 1 778 00:57:42,892 --> 00:57:45,332 2 to the 6 billionth is 1 779 00:57:45,332 --> 00:57:47,252 Because, okay, let's just go back 780 00:57:47,252 --> 00:57:49,272 2 to the 600 is 2 to the 6 781 00:57:49,272 --> 00:57:50,912 To the power of 100 782 00:57:50,912 --> 00:57:52,052 2 to the 6 is 1 783 00:57:52,052 --> 00:57:54,372 1 to the power of 100 is 1 784 00:57:54,372 --> 00:57:57,072 1 to the power of a billion is 1 785 00:57:57,072 --> 00:57:58,852 You want to hear something kind of funny? 786 00:57:58,972 --> 00:58:00,312 I'm going to expose 787 00:58:00,312 --> 00:58:02,992 Part of my trick of how I taught my children 788 00:58:02,992 --> 00:58:04,132 A lot of this math 789 00:58:04,132 --> 00:58:07,472 Especially when it comes to algebra 790 00:58:07,472 --> 00:58:16,552 I get so frustrated with them not understanding these certain generalities that I basically showed them that I could substitute. 791 00:58:17,032 --> 00:58:17,872 I don't need the number two. 792 00:58:17,952 --> 00:58:20,372 I could use a toilet seat as a variable. 793 00:58:21,512 --> 00:58:25,512 And I should draw a toilet seat and put it to, say, the sixth power. 794 00:58:25,632 --> 00:58:26,152 It's going to be one. 795 00:58:26,252 --> 00:58:28,432 Any number, you know, if it's in my finite field. 796 00:58:28,772 --> 00:58:35,712 And so, like, I call it toilet seat math where, like, it just doesn't matter anymore. 797 00:58:35,892 --> 00:58:37,152 Two to the sixth is one. 798 00:58:37,152 --> 00:58:42,292 And one to any power, one to the toilet seat, doesn't matter what it is, is going to be one. 799 00:58:42,872 --> 00:58:44,812 So here's the thing. 800 00:58:44,932 --> 00:58:47,812 So two to the 600 now we said is one. 801 00:58:47,912 --> 00:58:49,732 Two to the six billion is one. 802 00:58:49,892 --> 00:58:52,612 Two to the six gajillion is going to be one. 803 00:58:53,092 --> 00:58:59,552 Which really, you're starting to get to the point where, okay, remember what we were saying? 804 00:58:59,692 --> 00:59:01,572 Like, I can't even do this on a computer. 805 00:59:01,812 --> 00:59:02,792 I can't even do this math. 806 00:59:02,792 --> 00:59:09,972 But I know because of Fermat's little theorem that I can actually eliminate a large portion of this calculation. 807 00:59:10,452 --> 00:59:13,972 I could reduce it to 1 and discard it. 808 00:59:14,572 --> 00:59:14,652 Right. 809 00:59:15,332 --> 00:59:22,052 So you want to know what would be – if I asked you 2 to the 1,000th power, right? 810 00:59:22,752 --> 00:59:27,452 You would basically look at the 1,000 – you would say, well, what part of this can be reduced to 1? 811 00:59:27,452 --> 00:59:38,772 right so i know that like i know 960 is gonna be one so i got 40 left right so 36 out of that 40 812 00:59:38,772 --> 00:59:46,112 is gonna be one so i know that 996 i know two to the 996 is one which so if i wanted two to the 813 00:59:46,112 --> 00:59:51,772 1000th it's one times two to the fourth i know this is horrible to talk well this is well well 814 00:59:51,772 --> 00:59:56,152 what you're doing here which is just important is that you're taking your bit you're doing factoring 815 00:59:56,152 --> 01:00:00,612 which we haven't talked a lot on this podcast but anyone with you know i would say in elementary 816 01:00:00,612 --> 01:00:04,352 school math understands the concept of being able to break a number into smaller numbers 817 01:00:04,352 --> 01:00:12,312 um and like the number 100 you could also make 10 times 10 and what you're doing here in this 818 01:00:12,312 --> 01:00:18,212 exercise is you're breaking out each you're doing factoring but you're you're explicitly factoring 819 01:00:18,212 --> 01:00:24,992 out what can i factor out that goes to one because it's kind of just noise you're basically trying to 820 01:00:24,992 --> 01:00:29,132 you're removing everything that doesn't factor to one and then sitting in 821 01:00:29,132 --> 01:00:29,612 what's left. 822 01:00:30,292 --> 01:00:31,812 Which is in its own way, 823 01:00:31,852 --> 01:00:33,312 kind of what we've been doing with the modular arithmetic. 824 01:00:33,792 --> 01:00:33,832 Like, 825 01:00:34,232 --> 01:00:37,992 like what is a hundred module of seven? 826 01:00:38,092 --> 01:00:38,372 It's like, 827 01:00:38,412 --> 01:00:38,872 okay, 828 01:00:39,332 --> 01:00:40,292 well, 829 01:00:40,312 --> 01:00:45,452 I know 10 times seven is 70, 830 01:00:45,732 --> 01:00:45,852 right? 831 01:00:45,912 --> 01:00:48,932 And so you just start pulling these things out until you get what's left. 832 01:00:48,972 --> 01:00:49,332 And then you, 833 01:00:49,652 --> 01:00:51,492 instead of doing math on the entire number, 834 01:00:51,952 --> 01:00:54,632 you basically take out the garbage for what you can. 835 01:00:54,632 --> 01:00:57,552 that you know just goes to the identity property immediately 836 01:00:57,552 --> 01:00:59,752 and then stick with what's left to do the remaining math. 837 01:01:00,992 --> 01:01:05,332 So Fermat's little theorem powerfully exploits modular arithmetic 838 01:01:05,332 --> 01:01:09,092 to make a lot of this actually really easy. 839 01:01:09,272 --> 01:01:11,672 So it makes finding inverses formulaic. 840 01:01:11,772 --> 01:01:12,752 As long as it's prime. 841 01:01:13,012 --> 01:01:13,592 As long as the field is prime. 842 01:01:13,592 --> 01:01:14,952 As long as the field is prime. 843 01:01:15,572 --> 01:01:17,632 Which should light you up as like, 844 01:01:17,632 --> 01:01:24,292 oh, this is part of why my modulus in Bitcoin's elliptic curve 845 01:01:24,292 --> 01:01:26,132 is also a prime number. 846 01:01:26,532 --> 01:01:27,232 Correct? Yep. 847 01:01:29,632 --> 01:01:29,872 So 848 01:01:29,872 --> 01:01:30,932 that, you know, 849 01:01:31,572 --> 01:01:34,232 we're building, I feel like 850 01:01:34,232 --> 01:01:35,892 we're building a tower, right? 851 01:01:36,772 --> 01:01:38,172 So, you know, maybe the little 852 01:01:38,172 --> 01:01:40,252 parts of these bricks that we lay are a little painful, 853 01:01:40,492 --> 01:01:42,252 but we're building a tower to how do 854 01:01:42,252 --> 01:01:44,172 we know 855 01:01:44,172 --> 01:01:46,032 based on what we can see and visibly 856 01:01:46,032 --> 01:01:47,772 verify, right? 857 01:01:49,272 --> 01:01:50,192 How do we know Bitcoin 858 01:01:50,192 --> 01:01:51,772 works, right? 859 01:01:52,712 --> 01:01:54,012 We're building a big tower. 860 01:01:54,012 --> 01:01:58,672 For Matt's Little Theorem, I have to say, man, it's one of the big mind-blowing things. 861 01:01:58,772 --> 01:02:05,692 And I remember in Jimmy Song's Programming Bitcoin book, like seeing it being used but not like really explained. 862 01:02:05,892 --> 01:02:09,612 And I was like, oh, my God, how do they do this? 863 01:02:10,132 --> 01:02:15,532 You know, and it's like, okay, no, I got to – you go into some number theory textbooks and you just see it. 864 01:02:15,592 --> 01:02:16,432 It's like, oh, my God. 865 01:02:16,532 --> 01:02:19,192 This is one of the real beautiful, beautiful things. 866 01:02:19,512 --> 01:02:19,912 Yeah. 867 01:02:20,152 --> 01:02:23,672 I got some documents that – there's like many different ways to prove it. 868 01:02:23,672 --> 01:02:30,412 This is one of those things that people have walked into over the years because Fermat didn't like proving his stuff. 869 01:02:30,912 --> 01:02:33,032 But this one was easy. 870 01:02:33,312 --> 01:02:37,592 Maybe for the sake of the conversation today, we can explain what the exact equation is. 871 01:02:37,972 --> 01:02:42,132 And as a take-home exercise or maybe on a future episode, we can go into an actual proof of it. 872 01:02:42,232 --> 01:02:46,052 But if you have a field that is prime, let's call it P. 873 01:02:46,812 --> 01:02:47,392 Finite field. 874 01:02:48,112 --> 01:02:48,852 A finite field. 875 01:02:48,852 --> 01:02:51,792 If you have a finite field that's prime, let's just call that P for prime. 876 01:02:51,792 --> 01:03:04,592 the inverse of a number is that number to the p minus two modulo p yes and so let's take our 877 01:03:04,592 --> 01:03:11,072 example earlier um of a finite field of seven and you want to find the inverse of three well 878 01:03:11,072 --> 01:03:17,852 uh what you can wait can i just stop you for a second because i feel like i and i i'm just 879 01:03:17,852 --> 01:03:23,152 speaking from my own experience sure like where did p minus two just i thought you just said it 880 01:03:23,152 --> 01:03:28,692 was p minus one and you did this whole exercise to show it was that p minus one had this property 881 01:03:28,692 --> 01:03:34,332 why are we talking about p minus two so because without the benefit without the benefit of showing 882 01:03:34,332 --> 01:03:39,772 it on paper right yeah so the actual i i skipped a step of the actual theorem and i kind of 883 01:03:39,772 --> 01:03:49,112 rearrange things so to go one level higher um a so if a field is prime p a to the p minus one 884 01:03:49,112 --> 01:03:57,112 power equals one modulo p we just did that whole math we two to the six thousand we said two to the 885 01:03:57,112 --> 01:04:04,472 six is one three to the six is one any number inside that field modulo seven any number to the 886 01:04:04,472 --> 01:04:11,492 power of six is one. That's the first thing. That's the first step, right? Because now what 887 01:04:11,492 --> 01:04:16,712 we're trying to do is, okay, what are the two? Now, if I take a number in that field, 888 01:04:17,532 --> 01:04:21,692 right? Yes, I can raise it to the power of P minus one, but that's not what really we're 889 01:04:21,692 --> 01:04:25,312 trying to do now to find an inverse. We're going to use that property. We're going to leverage that 890 01:04:25,312 --> 01:04:31,592 property to find the inverse, right? We're going to say, what number do I multiply it by 891 01:04:31,592 --> 01:04:33,572 to get one. 892 01:04:34,112 --> 01:04:34,232 Right. 893 01:04:35,012 --> 01:04:37,312 And so to take it to the next level 894 01:04:37,312 --> 01:04:38,512 of the abstraction of the equation, 895 01:04:38,872 --> 01:04:41,912 if you're trying to look for the inverse of A 896 01:04:41,912 --> 01:04:43,332 in a prime field of P, 897 01:04:43,772 --> 01:04:48,392 the inverse of A equals A to the P minus two modulo P. 898 01:04:48,612 --> 01:04:50,592 And that's just basically rearranging the equation, 899 01:04:51,012 --> 01:04:53,032 which would be much easier to do 900 01:04:53,032 --> 01:04:54,432 with pen and paper and a visual, 901 01:04:54,592 --> 01:04:55,892 but just take my word for it. 902 01:04:55,952 --> 01:04:57,132 You can do it trivially yourself. 903 01:04:57,232 --> 01:04:58,432 You could rearrange the equation 904 01:04:58,432 --> 01:04:59,452 because what you're doing 905 01:04:59,452 --> 01:05:00,892 is you're basically trying to isolate 906 01:05:00,892 --> 01:05:06,992 the inverse of a uh so for the field seven which is a seven is a prime number 907 01:05:06,992 --> 01:05:12,232 and to plug this into the equation if you want to find the inverse of three 908 01:05:12,232 --> 01:05:19,672 so your three is a so now you have three to the p minus two which is seven minus two which is five 909 01:05:19,672 --> 01:05:27,672 so three to the fifth power is 243 equals three nine twenty seven eighty one two forty three 910 01:05:27,672 --> 01:05:37,312 bingo right and so 243 will mod 7 we're 243 equals 5 mod 7 yes and then so you do is yeah 911 01:05:37,312 --> 01:05:43,712 yeah and that so remember we we discussed this a half hour ago where we just said well three times 912 01:05:43,712 --> 01:05:49,152 five is 15 which means that those are those two are inverse those are associates right 913 01:05:49,152 --> 01:05:56,352 so like how do you use for my little theorem to get to that that's what basically rob just did 914 01:05:56,352 --> 01:06:06,612 right it took three to the power of five um what we have the benefit of like we've both been through 915 01:06:06,612 --> 01:06:11,892 a bunch of code and a bunch of textbooks that take this property for granted and i don't know 916 01:06:11,892 --> 01:06:18,812 about you rob but i have to be like how the heck do you just tell me that this number to the p minus 917 01:06:18,812 --> 01:06:26,752 two is my inverse. I see it implemented in Bitcoin books. And so it's because of this theorem, 918 01:06:27,072 --> 01:06:32,972 this property, right? And so he just demonstrated with the number three, let's maybe do an easier 919 01:06:32,972 --> 01:06:38,392 example. Remember we also said two and four are inverses of each other. So if I took two to the 920 01:06:38,392 --> 01:06:46,232 fifth power is 32, that's a lot easier, I think, to see that 32 mod seven is four. 921 01:06:46,232 --> 01:06:56,972 right 28 to 32 is four right 28 is four from it has a remainder right of four so that's just an 922 01:06:56,972 --> 01:07:02,272 easier way to um easier way to visualize it in our mind's eye which is what we're trying to do 923 01:07:02,272 --> 01:07:05,272 difficultly on this podcast of course yeah yeah 924 01:07:05,272 --> 01:07:12,552 um yes and i mean like it's funny because you could even go to a smaller prime field you could 925 01:07:12,552 --> 01:07:16,672 pick the prime field of three if you wanted to you can make these numbers smaller but with the 926 01:07:16,672 --> 01:07:21,792 core equation you can click and move these things oh and i gotta just say if it bothers you guys 927 01:07:21,792 --> 01:07:27,372 like i just if it bothers you right now that we just did this and it's not hasn't clicked that's 928 01:07:27,372 --> 01:07:31,832 fine i'm like i'm good with that because i'm hoping that just motivates you to just get a pen 929 01:07:31,832 --> 01:07:37,152 and paper and do some of these go to the study guy go through the examples and check it like it 930 01:07:37,152 --> 01:07:42,632 should bother you that's what that's it's like it should bother you as much as any point this is 931 01:07:42,632 --> 01:07:46,892 tripping you up you should be agitated and go figure it out a little bit right it's just the 932 01:07:46,892 --> 01:07:51,452 same way it should bother you that you don't know you don't maybe know if your private key is if 933 01:07:51,452 --> 01:07:57,092 your private key is invertible like it you don't really know until you actually learn this math it 934 01:07:57,092 --> 01:08:01,852 should that should bother you too yeah right and it should bother you a little bit that we're 935 01:08:01,852 --> 01:08:06,492 talking through this. But like, yeah, if you go through this exercise, two to the fifth, 936 01:08:06,712 --> 01:08:10,612 and why we say again, two to the fifth, I'm so sensitive to this because I've been through this 937 01:08:10,612 --> 01:08:19,172 so many times. Five is p minus two, right? It's a two. So you got, again, remember p to the n minus 938 01:08:19,172 --> 01:08:23,632 one, any number to the n minus one was one, right? So we're pulling that apart. And we're saying, 939 01:08:23,632 --> 01:08:31,132 well, we take that number times what? Well, that number times p minus two, remember how you add 940 01:08:31,132 --> 01:08:37,052 exponents when you multiply the same number by itself. So p minus 2 plus 1 is your p minus 1 941 01:08:37,052 --> 01:08:41,592 from Fermat's Little Theorem. So Fermat's Little Theorem doesn't give you the inverse. 942 01:08:41,972 --> 01:08:47,992 It gives you the property that allows you to sort of exploit that property to find that inverse. 943 01:08:48,332 --> 01:08:55,672 It's an extension. It's like an application. But any number in a finite field, modulo p, 944 01:08:56,192 --> 01:09:01,032 you can multiply by that number to the p minus 2, and boom, you get your inverse. And that's going 945 01:09:01,032 --> 01:09:05,952 to come in handy when you're multiplying by numbers that are bigger than atoms on this earth. 946 01:09:05,952 --> 01:09:22,288 Right Which I think to talk about that as a gateway and to directly touch a little bit a little bit of just how this works in Bitcoin is that we been using very small finite fields 7 17 3 whatever 947 01:09:22,948 --> 01:09:28,748 The universe of what we're talking about, though, in Bitcoin is 256 bits, right? 948 01:09:29,348 --> 01:09:32,948 256 zeros and ones and like the size of that universe. 949 01:09:32,948 --> 01:09:39,528 that would be really computationally a huge pain in the ass to do all the time 950 01:09:39,528 --> 01:09:45,448 but you could already see how unwieldy it gets if my prime is 17 right and i want to just say 951 01:09:45,448 --> 01:09:53,388 two to the 15th but i can tell you two to the 15th is going to multiply by two and give me the 952 01:09:53,388 --> 01:09:57,448 number of one in that field that much i know i know two to the 15th is my inverse of two 953 01:09:57,448 --> 01:10:00,748 I know that backwards and forwards 954 01:10:00,748 --> 01:10:02,708 because of Fermat's Little Theorem 955 01:10:02,708 --> 01:10:03,728 even though I 956 01:10:03,728 --> 01:10:05,908 do I know what 2 to the 15th is? 957 01:10:06,248 --> 01:10:06,568 no 958 01:10:06,568 --> 01:10:07,068 I think 959 01:10:07,068 --> 01:10:08,288 what is 2 to the 10th? 960 01:10:08,368 --> 01:10:08,888 1024? 961 01:10:09,148 --> 01:10:10,088 I can start doing it 962 01:10:10,088 --> 01:10:10,748 1024 963 01:10:10,748 --> 01:10:14,168 2048 is to the 11th 964 01:10:14,168 --> 01:10:16,248 4096 is to the 965 01:10:16,248 --> 01:10:16,708 like that's 966 01:10:16,708 --> 01:10:18,848 you already kind of see in your brain 967 01:10:18,848 --> 01:10:19,968 it's getting unwieldy 968 01:10:19,968 --> 01:10:22,348 and that's only the number 17 969 01:10:22,348 --> 01:10:23,388 okay right 970 01:10:23,388 --> 01:10:25,828 now you start dealing with computation 971 01:10:25,828 --> 01:10:26,828 that a computer can do 972 01:10:26,828 --> 01:10:31,408 It's going to get unwieldy at 2 to the 128th power. 973 01:10:32,048 --> 01:10:32,428 Right. 974 01:10:32,608 --> 01:10:34,208 It's going to start to get unwieldy. 975 01:10:35,108 --> 01:10:35,248 Yeah. 976 01:10:35,328 --> 01:10:35,528 Right? 977 01:10:36,068 --> 01:10:36,488 Exactly. 978 01:10:37,108 --> 01:10:41,528 And so there's ways that you can further kind of chunk this up, so to speak. 979 01:10:41,828 --> 01:10:43,608 So it's something that computers can do. 980 01:10:43,968 --> 01:10:48,688 And this is also really important from a security standpoint. 981 01:10:48,688 --> 01:10:59,128 The reason why Satoshi started with 500 lines of the LibSecP, well, of the LibSecP part of the code base, and it turned, well, the SecP operations. 982 01:11:00,588 --> 01:11:06,628 Avid podcast, Magic Internet Math superfan, Richard Hart. 983 01:11:07,208 --> 01:11:08,468 Yes, yes he is. 984 01:11:08,468 --> 01:11:15,268 Last two weeks ago, did a very correct technicality call out of what I said. 985 01:11:15,268 --> 01:11:21,508 I was being very short and fast with my explanation of the SecP 256k1 curve. 986 01:11:21,888 --> 01:11:25,528 That is not the LibSecP library Bitcoin uses. 987 01:11:25,768 --> 01:11:27,888 The LibSecP library is kind of think of it. 988 01:11:28,008 --> 01:11:34,008 If SecP 256k1 is like this theoretical like field that we use all of our public private 989 01:11:34,008 --> 01:11:39,408 key math and Bitcoin around, LibSecP is the execution and implementation of how you work 990 01:11:39,408 --> 01:11:39,748 with that. 991 01:11:40,008 --> 01:11:42,508 And that actually didn't come to exist originally in Bitcoin. 992 01:11:42,508 --> 01:11:48,888 um satoshi was pulling in i think from like an open ssl library originally and that has problems 993 01:11:48,888 --> 01:11:55,768 and concerns because the level of care you would put for securing web traffic is different than if 994 01:11:55,768 --> 01:12:04,688 that library was securing money there's a different threat model involved and so the problem is is 995 01:12:04,688 --> 01:12:15,968 that you need to compute this like a to the p minus 2 mod p but p instead of being 7 is 2 to the 996 01:12:15,968 --> 01:12:25,968 256 not 256 2 to the 256 so 2 times 2 times 2 times 2 and you get this number that's bigger than 997 01:12:25,968 --> 01:12:31,388 is it 2 to the 256 minus 2 to the 32 minus 977 yes so i did a little bit of a shortcut there 998 01:12:31,388 --> 01:12:33,668 It's 2 to the 256 minus 2 to the 32. 999 01:12:34,028 --> 01:12:38,528 Just saving us from having to explain another correction next week. 1000 01:12:38,708 --> 01:12:43,268 We talked about this last episode. 1001 01:12:43,428 --> 01:12:49,008 It's 2 to the 256 minus 2 to the 232 minus 2 to the 9 minus 2 to the 8 minus 2 to the 7 minus 2 to the 6 minus 2 to the 4 minus 1. 1002 01:12:49,008 --> 01:12:50,148 I think that's 977. 1003 01:12:50,888 --> 01:12:53,168 That last piece is 977, I think. 1004 01:12:53,548 --> 01:12:54,888 So there's 77 digits. 1005 01:12:55,208 --> 01:13:00,808 The number has 115-79208-92008, and it goes on for 77 digits. 1006 01:13:00,808 --> 01:13:06,408 But the distinction, though, is because a general hash function only uses 2 to the 256, right? 1007 01:13:07,188 --> 01:13:09,308 Yeah, and that's exactly right. 1008 01:13:09,428 --> 01:13:14,448 So when you're doing these operations, they're kind of constrained and bound within that. 1009 01:13:15,708 --> 01:13:21,828 By the way, I got another correction I should just mention while we're doing the errata section of the podcast. 1010 01:13:22,308 --> 01:13:22,528 Yes. 1011 01:13:22,768 --> 01:13:27,148 Because I posted the video series to Stacker News and somebody gave it – 1012 01:13:27,148 --> 01:13:30,788 Dude, shout out to fucking people who give these corrections, by the way. 1013 01:13:30,928 --> 01:13:31,528 Shout out Richard. 1014 01:13:31,528 --> 01:13:32,328 I wish I had your username. 1015 01:13:32,588 --> 01:13:33,428 Shout out Richard Hart. 1016 01:13:34,228 --> 01:13:42,468 But also, this is like, dude, this is like Hart talking about, Rob and I are not, we're 1017 01:13:42,468 --> 01:13:46,848 trying to create a conversation and we'll miss some things and maybe, so I think the 1018 01:13:46,848 --> 01:13:51,388 correction I got on Stacker News was the, and it wasn't on the podcast, it was on the 1019 01:13:51,388 --> 01:13:55,988 video, but the video is based on everything the podcast is based on, was that the K and 1020 01:13:55,988 --> 01:14:03,788 the k meant cobblitz curve in sec p which it was a very fine distinction and i'll be honest i don't 1021 01:14:03,788 --> 01:14:11,128 fully understand the distinction but there is a distinction here with the k in 256 k1 1022 01:14:11,128 --> 01:14:16,968 not exactly meaning that it's a cobblitz not necessarily exactly meaning it's a cobblitz 1023 01:14:16,968 --> 01:14:21,448 curve this is something i was i don't even know if the correction is true but i was corrected but 1024 01:14:21,448 --> 01:14:28,788 But the correction was triangulated by another Claude call somewhere down the line. 1025 01:14:28,948 --> 01:14:30,808 And I was like, oh, that guy's probably right. 1026 01:14:31,388 --> 01:14:31,528 Yeah. 1027 01:14:32,268 --> 01:14:32,708 Yeah. 1028 01:14:32,968 --> 01:14:35,128 No, don't trust Verify, including us guys. 1029 01:14:36,548 --> 01:14:36,768 Yeah. 1030 01:14:37,228 --> 01:14:42,348 So for the moment, before I make a bunch more of additional errors, we'll conclude the errata section. 1031 01:14:42,468 --> 01:14:44,708 But we'll do another one in about 90 seconds. 1032 01:14:46,288 --> 01:14:49,628 So you have this really big number, which I just fully listed out before. 1033 01:14:49,628 --> 01:14:53,028 2 to the 56 minus 2 to the 32 minus 2 to the 9, whatever, right? 1034 01:14:53,088 --> 01:14:53,768 We need a good word. 1035 01:14:53,828 --> 01:14:56,108 We need to need a good word for this. 1036 01:14:56,188 --> 01:14:57,208 It's the Bitcoin number. 1037 01:14:58,048 --> 01:14:59,428 It's the Bitcoin Prime. 1038 01:14:59,908 --> 01:15:00,768 We call it the Bitcoin Prime. 1039 01:15:00,948 --> 01:15:01,748 It's the Bitcoin Prime. 1040 01:15:01,988 --> 01:15:07,928 That's a great name for a slop account that does news, the Bitcoin Prime. 1041 01:15:08,388 --> 01:15:09,128 The Bitcoin Prime. 1042 01:15:09,208 --> 01:15:09,628 I like it. 1043 01:15:09,748 --> 01:15:10,208 I like that. 1044 01:15:10,568 --> 01:15:10,708 Yeah. 1045 01:15:11,868 --> 01:15:17,868 So your finite modular prime field is this massive number. 1046 01:15:17,868 --> 01:15:22,748 one that you're not going to be we're not going to try and manually calculate one of these out but 1047 01:15:22,748 --> 01:15:29,888 what you can do it's really interesting um is that it's a binary number right so if you have zero 1048 01:15:29,888 --> 01:15:37,988 and one and then you just each each additional digit in binary is a new power of two so zero 1049 01:15:37,988 --> 01:15:45,768 right is just zero one you could have as the number one but then you add the next number so 1050 01:15:45,768 --> 01:15:51,488 So now you have one zero in binary is the number two. 1051 01:15:52,508 --> 01:15:55,088 The number one one in binary is the number three. 1052 01:15:55,088 --> 01:16:01,128 Yeah, let me also just mention here, if you've gotten this far, 1053 01:16:01,748 --> 01:16:06,928 that on my website, on magicinternetmath.com, there is a game. 1054 01:16:07,448 --> 01:16:10,268 There's two games, actually, that I think are really cool and useful, 1055 01:16:10,608 --> 01:16:11,208 and you should go. 1056 01:16:11,668 --> 01:16:12,928 There are four skill building. 1057 01:16:13,088 --> 01:16:14,968 One is called Modular. 1058 01:16:14,968 --> 01:16:21,508 it's a racing game doing modular arithmetic and um it progressively gets harder with like we just 1059 01:16:21,508 --> 01:16:26,468 did like the first level is like mod five and then he goes to like mod 17 and then maybe mod bigger 1060 01:16:26,468 --> 01:16:31,488 numbers but like um so it's like you can play against computer play against your friends and 1061 01:16:31,488 --> 01:16:37,648 get points and build skill doing modular arithmetic and another one is base i'm doing it's conversion 1062 01:16:37,648 --> 01:16:44,108 of binary decimal and hex and i think at the design the thinking there is just get this would 1063 01:16:44,108 --> 01:16:49,088 be a lot more fluid and easier conversation if we didn't have to like explain binary arithmetic 1064 01:16:49,088 --> 01:16:57,568 all the time so like it'd be good to know that like just off the top of your head 10 10 101 0 1065 01:16:57,568 --> 01:17:04,748 is the number uh is number 10 right it's just like 101 1 is number 11 decimal and like i think 1066 01:17:04,748 --> 01:17:10,268 it's just helpful if you do enough of these and i i made the game for myself like i remember 1067 01:17:10,268 --> 01:17:15,308 of coding this in Florida and on the flight home, all I did was play this game over and over again, 1068 01:17:15,788 --> 01:17:21,648 doing these binary conversions. And it's very helpful to see a four number binary and know 1069 01:17:21,648 --> 01:17:28,788 exactly what number between zero and 16 or zero and 15 it is. Yeah. And so this is to tie that, 1070 01:17:28,788 --> 01:17:34,268 I think being fluid in all of these different units that you do operations in is pretty important. 1071 01:17:34,268 --> 01:17:42,488 And the way I would tee this up to make this like a problem, because I started going down binary, which is a different representation of numbers. 1072 01:17:42,608 --> 01:17:47,108 But like what's actually really interesting here, let's say if you wanted to do 3 to the power of 8. 1073 01:17:47,568 --> 01:17:50,548 3 times 3 times 3 times 3 times 3 times 3 times 3. 1074 01:17:50,668 --> 01:17:51,368 You do it 8 times. 1075 01:17:51,428 --> 01:17:55,968 And I'm keeping it as a small number to explain when you get to really, really, really, really big numbers. 1076 01:17:56,488 --> 01:17:57,328 64.01? 1077 01:17:57,868 --> 01:17:59,448 Right, like how big it can go. 1078 01:17:59,468 --> 01:18:00,008 That's my guess. 1079 01:18:00,208 --> 01:18:00,788 That's just my guess. 1080 01:18:00,788 --> 01:18:01,248 64.01. 1081 01:18:01,248 --> 01:18:02,608 65.61. 1082 01:18:02,808 --> 01:18:03,268 6,561. 1083 01:18:03,268 --> 01:18:07,728 561. But what you can do though is a shortcut instead of doing three times three times seven 1084 01:18:07,728 --> 01:18:12,308 times three, just start grouping them together and squaring them. Three times three is nine. Cool. 1085 01:18:12,748 --> 01:18:17,948 And then you have nine times nine is 81. I tried doing 81 squared. You just did three to the four. 1086 01:18:18,688 --> 01:18:25,108 Yeah, I tried doing 81 squared in my head and I failed. Yeah, so 81 squared is 6,561. So instead 1087 01:18:25,108 --> 01:18:30,448 of you can do three squarings instead of seven multiplications. And it's like three versus seven, 1088 01:18:30,448 --> 01:18:33,028 That's not that big of a deal, but we're still dealing with small numbers. 1089 01:18:33,128 --> 01:18:41,588 When you start getting to the number of just actual, like if you did it at these large, like the Bitcoin prime number, how unruly it gets quickly. 1090 01:18:41,708 --> 01:18:42,868 Well, here's the power of that, right? 1091 01:18:42,868 --> 01:18:52,928 If I said you're doing mod 10, right, and you said what's 3 to the 8th power mod 10, then I already know 81 is 3 to the 4th, and that's 1. 1092 01:18:53,128 --> 01:18:55,908 So again, that's like my – it's not Fermat's little theorem. 1093 01:18:55,908 --> 01:19:02,048 This goes into – it turns out that Euler generalized Fermat's theorem if your modulo is not a prime. 1094 01:19:03,668 --> 01:19:04,868 And it's brilliant. 1095 01:19:05,228 --> 01:19:09,148 And I have videos buried in my – it's got four views. 1096 01:19:10,748 --> 01:19:24,168 But it's a really cool video of me actually sitting there and doing all that math and showing you how Euler generalized with the Toshin function, generalized Fermat's little theorem for – if your modulo is not prime. 1097 01:19:24,168 --> 01:19:30,208 then your finite you're sort of like what your analogy to the finite field is now it's it's a 1098 01:19:30,208 --> 01:19:34,788 group of units that you know has inverses well this is where the grouping gets really interesting 1099 01:19:34,788 --> 01:19:40,168 right so we're getting to this like bitcoin prime number which for the sake of conversation it's not 1100 01:19:40,168 --> 01:19:46,268 two to the 256 but you're doing two to the 256 minus two to the 232 which is infinitesimally 1101 01:19:46,268 --> 01:19:52,608 smaller comparatively it's effectively in the scope of being similar to two to the 256 because 1102 01:19:52,608 --> 01:19:56,688 the things you're subtracting away are just infinitesimally small compared to the scale of it 1103 01:19:56,688 --> 01:20:03,788 um what's interesting about this is that it very nicely starts collapsing on itself if you are 1104 01:20:03,788 --> 01:20:09,488 doing the squaring trick that i explained where instead of square multiply yeah well yeah exactly 1105 01:20:09,488 --> 01:20:16,148 instead of doing two to the 256 multiplication operations you can square and multiply 512 times 1106 01:20:16,148 --> 01:20:20,848 and that's something that a computer can do 512 mathematical operations trivially fast 1107 01:20:20,848 --> 01:20:27,208 yeah by the way i'm really struggling right now because i want to i feel like i want to rename the 1108 01:20:27,208 --> 01:20:31,968 podcast the bitcoin prime that's just a whole nother whole nother issue that i'm dealing with 1109 01:20:31,968 --> 01:20:37,808 in the back of my head right now so okay anyway um i struggled with this algorithm square multiplier 1110 01:20:37,808 --> 01:20:43,188 a little bit except the only way i kind of like reconcile it for myself is when i think of um 1111 01:20:43,188 --> 01:20:49,808 by the way this is all this is all euclidean algorithm everything we were saying before 1112 01:20:49,808 --> 01:20:55,948 It's just a way of unpackaging and repackaging the process of the long division. 1113 01:20:56,668 --> 01:21:03,028 What makes sense to me – like I always struggle with bit shifting except when I think of it in decimal and multiplying. 1114 01:21:03,588 --> 01:21:07,188 I always know that multiplying by 10 adds a zero, right? 1115 01:21:07,768 --> 01:21:08,008 Right. 1116 01:21:08,048 --> 01:21:10,808 Well, in binary, multiplying by 2 adds a zero. 1117 01:21:11,028 --> 01:21:17,108 This is like – that's the only way I reconcile that in my head is like that multiplying by the base is what adds a zero. 1118 01:21:17,108 --> 01:21:17,588 Right. 1119 01:21:17,588 --> 01:21:27,648 Right. Well, this is where I started with binary. The reason why is because this basically the square and multiply trick is leveraging binary representation. 1120 01:21:28,648 --> 01:21:45,868 And this is something we also, I think, briefly touched upon last episode is that the reason why the LibSecP library exists and we're not using Satoshi's original code is that when the math is actually the money, you have to take a different security posture than if I'm securing my website with an open SSL library. 1121 01:21:46,788 --> 01:21:50,948 One of the ways you can attack a system is called a timing analysis attack. 1122 01:21:51,468 --> 01:22:07,128 And so if you are able to, if you're sitting on a computer and you're able to see the CPU usage and you're able to see exactly how it's spiking and how it's computing all of these numbers, you can start inferring what the actual number is. 1123 01:22:07,168 --> 01:22:09,688 And if you know the number, you know the private key, right? 1124 01:22:09,688 --> 01:22:15,148 and so the way the libsec p library handles this the reason why i was going with binary is because 1125 01:22:15,148 --> 01:22:19,828 like you said every time you multiply by 10 you add a zero at the end every time you multiply by 1126 01:22:19,828 --> 01:22:24,628 two you add a zero at the end if you're doing binary representation you can take this number 1127 01:22:24,628 --> 01:22:35,388 as 256 zeros and ones and you can just work on each of those um the way libsec p 256 k1 gets 1128 01:22:35,388 --> 01:22:40,068 around this is that no matter what the number is it oh it basically does a bunch of extra garbage 1129 01:22:40,068 --> 01:22:45,048 math so that it looks like it's taking the same amount of time no matter what the number is yeah 1130 01:22:45,048 --> 01:22:51,268 let me ask you let me try something here this this notion of a timing attack is the equivalent 1131 01:22:51,268 --> 01:22:57,608 in cryptography of like knowing that the word the is a very common three-letter word 1132 01:22:57,608 --> 01:23:03,008 and being able to say well my i don't have to brute force everything here because i can actually 1133 01:23:03,008 --> 01:23:08,988 i know that e is the most popular letter and i know that i can try to substitute that and get a 1134 01:23:08,988 --> 01:23:13,848 good clue as to how i can eliminate a lot of them a lot of the brute force math just by knowing those 1135 01:23:13,848 --> 01:23:21,348 facts so a timing attack is sort of a similar way of yeah reducing the computation it takes to to 1136 01:23:21,348 --> 01:23:26,948 if because you're giving those clues i'll do a rare throw out to pop culture here and if anyone's 1137 01:23:26,948 --> 01:23:35,088 familiar with the movie uh like the enigma like enigma with the basically alan turing and the 1138 01:23:35,088 --> 01:23:40,288 basically the war effort to decrypt all of the nazi communication in the movie they have like 1139 01:23:40,288 --> 01:23:46,128 the aha movement where they realize every single cable that the axis powers are sending starts with 1140 01:23:46,128 --> 01:23:52,768 heil hitler and because it says that they reverse engineer and say wait a second we know for a fact 1141 01:23:52,768 --> 01:23:57,308 they're always starting with this and they use that as a way to start deconstructing and 1142 01:23:57,308 --> 01:24:02,188 understanding the rest of the message that's being said and because of that they're able to 1143 01:24:02,188 --> 01:24:07,708 basically break down the entire thing that's a version of like in a way a timing attack yes so 1144 01:24:07,708 --> 01:24:15,788 that's that's really it so it's the ability to essentially you know completely reduce a lot of 1145 01:24:15,788 --> 01:24:21,068 the computation that would otherwise be brute force because you've introduced a pattern yeah 1146 01:24:21,068 --> 01:24:25,648 and the right i'm realizing i'm waiting for the southern power of the law center to put me on 1147 01:24:25,648 --> 01:24:29,888 their hate list because i just said that on a podcast like no you know what you've done i 1148 01:24:29,888 --> 01:24:35,048 this is not to get too inside this podcast is getting deep monetized now no losing my career 1149 01:24:35,048 --> 01:24:40,868 no not at all it's the other way around i've done this before because you know i'm a jewish guy and 1150 01:24:40,868 --> 01:24:47,048 i talk about this stuff sometimes this podcast is over well we i have my some of my podcast 1151 01:24:47,048 --> 01:24:54,028 partners have a theory that's like i've triggered the massad um machine to to and so because we've 1152 01:24:54,028 --> 01:24:58,108 seen some spikes and downloads in certain episodes where we ended up talking about that and we're 1153 01:24:58,108 --> 01:25:03,168 like what the hell why is this happening so um you might have you might have sparked i might i might 1154 01:25:03,168 --> 01:25:07,808 have got a spike in downloads you might have hit the aggregator on this one yeah for the cia massad 1155 01:25:07,808 --> 01:25:12,128 everyone listening send some boosts especially right now right i mean although it'd be hard to 1156 01:25:12,128 --> 01:25:18,588 it's it'd be hard to be noticeable now in um but we this is like two days after the iran attack and 1157 01:25:18,588 --> 01:25:23,628 everybody hates um jewish people in the world because of it i think they did before the two 1158 01:25:23,628 --> 01:25:29,648 days ago but also now they didn't yeah but now it's like well the noticing is definitely up there 1159 01:25:29,648 --> 01:25:34,268 right now and uh so they might be they probably extended the podcast i think you might have helped 1160 01:25:34,268 --> 01:25:38,888 the download count there you go perfect great but anyway that's the example of a timing attack right 1161 01:25:38,888 --> 01:25:42,488 you have you have fingerprinting that you're able to always determine a single look at and be like 1162 01:25:42,488 --> 01:25:45,688 oh if you're always starting with this or your point like e is the most common letter right 1163 01:25:45,688 --> 01:25:50,328 and with the way the lipstick library the lipstick p56k1 library ultimately handles this 1164 01:25:50,328 --> 01:25:56,128 is it adds a bunch of noise and so that every the computation for every number on the elliptic curve 1165 01:25:56,128 --> 01:26:02,728 looks like the same amount of computation no matter what and that was not part of the original 1166 01:26:02,728 --> 01:26:07,248 library because they were using the open ssl library to secure websites and stuff it wasn't 1167 01:26:07,248 --> 01:26:12,168 securing the money but this is the stuff that everyone leverages and uses when they're actually 1168 01:26:12,168 --> 01:26:20,248 using bitcoin most hardware wallets import this library as how they do their signature operations 1169 01:26:20,248 --> 01:26:24,448 this is another example of why you don't want to roll your own cryptography most of the time 1170 01:26:24,448 --> 01:26:28,328 because the moment you're like oh i found a shortcut it does it even faster and then you 1171 01:26:28,328 --> 01:26:32,408 realize like you're not doing constant time computation and anyone who's sitting on your 1172 01:26:32,408 --> 01:26:36,808 machine can infer what your number is is like not a great spot to be in yeah so it's worth 1173 01:26:36,808 --> 01:26:41,248 first probably worth glazing because we want to get these guys in the podcast it's worth glazing 1174 01:26:41,248 --> 01:26:48,508 glazing tim ruffing and uh jonas nick a little bit and the like you know it's a double-edged 1175 01:26:48,508 --> 01:26:54,268 sword we talk about here because you know one of the reasons we've all gotten away with not being 1176 01:26:54,268 --> 01:26:58,768 karens at the bitcoin store is because these guys did all the work these guys are the managers and 1177 01:26:58,768 --> 01:27:04,648 they're doing good they did the good work already to make sure that we didn't have to ask these 1178 01:27:04,648 --> 01:27:09,488 questions, even though what Rob and I are doing now is saying, well, let's back up a 1179 01:27:09,488 --> 01:27:09,988 second, man. 1180 01:27:10,888 --> 01:27:15,248 These guys have done good work, but are we always going to live in that world where those 1181 01:27:15,248 --> 01:27:16,308 people are doing good work? 1182 01:27:16,308 --> 01:27:23,508 I mean, we are adversarial by nature, and we should be, and we should have the ability 1183 01:27:23,508 --> 01:27:30,188 to know when to glaze these guys and say good work because we can see what they've done 1184 01:27:30,188 --> 01:27:34,088 and understand it versus saying thank God for being benevolent dictators. 1185 01:27:34,648 --> 01:27:56,548 Exactly. Exactly. Right. So like, I want to point out, I want to make sure that that's clear and that we, you know, again, having the agency to say good, good call guys. Like, that's the kind of thing you only get at the end of the suffering through the end of this podcast. This is a nice job, Tim Ruffing. 1186 01:27:56,548 --> 01:28:04,368 yeah right yeah especially like we're i think we're getting to a point where 1187 01:28:04,368 --> 01:28:10,828 i think we've just about gone through all of chapter two we did a little bit of a sprinkle 1188 01:28:10,828 --> 01:28:15,168 uh candy at the end for those that wanted to understand how it works with lipsec p the 1189 01:28:15,168 --> 01:28:23,908 library itself in bitcoin um there's every single step of this library you could do a whole hour 1190 01:28:23,908 --> 01:28:28,028 and a half long podcast talking about the nuances and how they put the thought into it and it's a 1191 01:28:28,028 --> 01:28:34,288 culmination of a decade of work um we covered a lot of chapter three two here but we what we've 1192 01:28:34,288 --> 01:28:38,968 stopped where we've stopped short where we'll probably pick up next time if we don't have if 1193 01:28:38,968 --> 01:28:45,188 we don't have our guest already where we stop short is the discrete log problem right would 1194 01:28:45,188 --> 01:28:53,068 the discrete log problem be a good spot to do solo so we're not bringing someone to do that 1195 01:28:53,068 --> 01:28:54,428 and then we can tee it up. 1196 01:28:55,328 --> 01:28:57,008 I'm going to be at Op Next 1197 01:28:57,008 --> 01:28:59,808 since now March 1st. 1198 01:29:00,188 --> 01:29:03,108 And Jonas Nick will also be presenting there. 1199 01:29:03,988 --> 01:29:06,468 We should try and just do a live rip. 1200 01:29:06,888 --> 01:29:08,868 That's a good idea. 1201 01:29:09,568 --> 01:29:11,568 Let's see if we can finagle that. 1202 01:29:11,948 --> 01:29:13,368 Op Next is April. 1203 01:29:14,328 --> 01:29:15,328 I'm looking at it right here. 1204 01:29:15,588 --> 01:29:16,548 15th or something like that. 1205 01:29:16,648 --> 01:29:17,368 It's in the middle of April. 1206 01:29:17,868 --> 01:29:18,488 16th, okay. 1207 01:29:18,568 --> 01:29:19,208 April 16th. 1208 01:29:19,208 --> 01:29:20,388 Thursday, April 16th. 1209 01:29:20,388 --> 01:29:23,128 I'll be in town in New York City 1210 01:29:23,128 --> 01:29:24,208 if you're going to be there 1211 01:29:24,208 --> 01:29:25,148 absolutely 1212 01:29:25,148 --> 01:29:27,948 and that would actually give us 1213 01:29:27,948 --> 01:29:30,308 one more, I'm looking at the date 1214 01:29:30,308 --> 01:29:33,028 that would actually give us two more podcasts before we do a live show with him 1215 01:29:33,028 --> 01:29:34,688 so if we could actually figure out what we do 1216 01:29:34,688 --> 01:29:36,708 if we get his scheduled permit 1217 01:29:36,708 --> 01:29:38,848 and work for it, he said he was interested in coming on 1218 01:29:38,848 --> 01:29:40,908 but I didn't realize that 1219 01:29:40,908 --> 01:29:42,348 if we're all going to be in the same room 1220 01:29:42,348 --> 01:29:44,928 it'd be really cool to just rip for 90 minutes 1221 01:29:44,928 --> 01:29:46,968 and maybe you and I can do a pilot 1222 01:29:46,968 --> 01:29:49,128 live rip when we're in Nashville 1223 01:29:49,128 --> 01:29:49,968 when I'm in Nashville 1224 01:29:49,968 --> 01:29:52,328 because we don't know. 1225 01:29:52,508 --> 01:29:54,448 I do five podcasts. 1226 01:29:54,608 --> 01:29:56,028 I don't do a lot of live rips. 1227 01:29:56,128 --> 01:29:59,488 I don't even know if I have the technical ability right now 1228 01:29:59,488 --> 01:30:00,608 so we can prepare for that. 1229 01:30:01,088 --> 01:30:02,908 Yes, I would like that a lot. 1230 01:30:03,568 --> 01:30:04,368 That'd be great. 1231 01:30:05,488 --> 01:30:08,408 And I should be getting just back to the National 1232 01:30:08,408 --> 01:30:10,228 for Grassroots Bitcoin just as that's happening. 1233 01:30:10,728 --> 01:30:11,788 God, what a month. 1234 01:30:12,088 --> 01:30:14,548 April's sick because we'll be in Nashville, 1235 01:30:14,948 --> 01:30:16,628 we'll have Opnex, and then we'll be in Vegas. 1236 01:30:17,148 --> 01:30:18,408 Yeah, it's going to be a busy... 1237 01:30:18,408 --> 01:30:25,948 maybe we'll do three live rips yeah and bitcoin plus plus is happening before vegas um i maybe 1238 01:30:25,948 --> 01:30:31,188 and then there's a fish show we're gonna start there's a fish show there might be psychedelics 1239 01:30:31,188 --> 01:30:36,028 and math and combining everything together yeah this might be this might get out of control really 1240 01:30:36,028 --> 01:30:42,648 quick but um yeah dude this was cool this is great i like this banter a little bit at the end 1241 01:30:42,648 --> 01:30:46,868 because we just need to kind of like land, just land the ship a little bit. 1242 01:30:46,868 --> 01:30:52,088 We just went through, I think a lot of people just went through something kind of difficult 1243 01:30:52,088 --> 01:30:54,748 and possibly injurious. 1244 01:30:55,788 --> 01:30:56,148 Absolutely. 1245 01:30:56,608 --> 01:30:59,728 And the reality is we are just, it's baby steps. 1246 01:31:00,148 --> 01:31:04,408 We have a large driveway to shovel and it's baby steps. 1247 01:31:04,868 --> 01:31:09,448 And we just want good form and understand where we're going. 1248 01:31:09,448 --> 01:31:32,368 Whether you like the analogy of knitting a sweater or you like the analogy of navigating the iceberg, this is about developing an ethic of spending a little bit of time every day on this and eventually building something substantial, substantial bedrock of your foundation. 1249 01:31:33,268 --> 01:31:33,668 Agreed. 1250 01:31:33,668 --> 01:31:38,328 Yeah, just listen to the podcast, go through the study guide. 1251 01:31:38,328 --> 01:31:49,208 I think the once every other week cadence for these episodes is a good way to like find an hour in every two weeks just to flip through this stuff and then listen to the podcast. 1252 01:31:49,368 --> 01:31:50,608 Like it's a good way to, 1253 01:31:51,208 --> 01:31:51,408 you know, 1254 01:31:51,428 --> 01:31:54,448 over the course of a year and if you find a podcast later, 1255 01:31:54,468 --> 01:31:56,568 just start doing that rough cadence. 1256 01:31:56,728 --> 01:31:58,608 So you get a chance to retain information, 1257 01:31:58,768 --> 01:31:59,648 think about it. 1258 01:32:00,968 --> 01:32:01,368 Yeah. 1259 01:32:01,468 --> 01:32:06,408 Trying to provide a good way for people to get as like a ladder to get brought up into this. 1260 01:32:06,828 --> 01:32:07,388 That's right. 1261 01:32:07,388 --> 01:32:12,948 So just keep grinding, keep climbing, and we will see you in two weeks. 1262 01:32:13,188 --> 01:32:13,668 Two weeks. 1263 01:32:13,768 --> 01:32:13,928 Peace.