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Errors:
*Typo on the «hard problem» at 14:11, it should be a/(b c) b/(a c) c/(a b) = 4
*Typo-turned-speako: The classification of quasithin groups is 1221 pages long, not 12,000. The full collection of papers proving the CFSG theorem do make up tens of thousands of pages, but no one paper was quite _that_ crazy.
Thanks to Richard Borcherds for helpful comments while putting this video together. He has a wonderful hidden gem of a channel: youtu.be/a9k_QmZbwX8
The Monster image comes from the Noun Project, via Nicky Knicky
— These animations are largely made using manim, a scrappy open-source python library: github.com/3b1b/manim
If you want to check it out, I feel compelled to warn you that its not the most well-documented tool, and it has many other quirks you might expect in a library someone wrote with only their own use in mind.
If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then «add subtitles/cc». I really appreciate those who do this, as it helps make the lessons accessible to more people.
— 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted on new videos, subscribe: 3b1b.co/subscribe
How do you secure messages over the internet? How do quantum computers break it? How do you fix it? Why dont you watch the video to find out? Why does this description have so many questions? Why are you still reading? What is the meaning of life?
0:00 Intro — Are we DOOOOMED??
0:52 How NOT to Send Secret Messages
2:09 RSA — Encryption Today
5:19 One-Way Functions and Post-Quantum Cryptography
7:28 Qubits and Measurement
9:03 BB84 — Quantum Cryptography
12:43 Alternatives and Problems
14:26 A Case for Quantum Computing
CLARIFICATIONS:
You dont actually need a quantum computer to do quantum-safe encryption. As briefly mentioned at 7.04, there are encryption schemes that can be run on regular computers that cant be broken by quantum computers.
CORRECTIONS:
«The public key can only be used to scramble information.» (2.18) Technically, you can use any key to encrypt or decrypt whatever you want. But theres a specific way to use them thats useful, which is whats shown in the video.
«Given a private key, its easy to create its corresponding public key.» (5.36) In RSA, depending on exactly what you mean by «private key», neither key is actually derivable from the other. When they are created, they are generated together from a common base (not just the public key from the private key). But typically, the file that stores the «private key» actually contains a bit more information than just the private key. For example, in PKCS #1 RSA private key format ( tools.ietf.org/html/rfc3447#appendix-A.1.2 ), the file technically contains the entire public key too. So in short, you technically cant get the public key from the private key or vice versa, but the file that contains the private key can hold more than just the private key alone, making it possible to retrieve the public key from it.
Important error correction: In the video, I say that Dirichlet showed that the primes are equally distributed among allowable residue classes, but this is not historically accurate. (By «allowable», here, I mean a residue class whose elements are coprime to the modulus, as described in the video). What he actually showed is that the sum of the reciprocals of all primes in a given allowable residue class diverges, which proves that there are infinitely many primes in such a sequence.
Dirichlet observed this equal distribution numerically and noted this in his paper, but it wasnt until decades later that this fact was properly proved, as it required building on some of the work of Riemann in his famous 1859 paper. If Im not mistaken, I think it wasnt until Vallée Poussin in (1899), with a version of the prime number theorem for residue classes like this, but I could be wrong there.
In many ways, this was a very silly error for me to have let through. It is true that this result was proven with heavy use of complex analysis, and in fact, its in a complex analysis lecture that I remember first learning about it. But of course, this would have to have happened after Dirichlet because it would have to have happened after Riemann!
My apologies for the mistake. If you notice factual errors in videos that are not already mentioned in the videos description or pinned comment, dont hesitate to let me know.
— These animations are largely made using manim, a scrappy open-source python library: github.com/3b1b/manim
If you want to check it out, I feel compelled to warn you that its not the most well-documented tool, and it has many other quirks you might expect in a library someone wrote with only their own use in mind.
If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then «add subtitles/cc». I really appreciate those who do this, as it helps make the lessons accessible to more people.
— 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted on new videos, subscribe: 3b1b.co/subscribe
Typo correction: At 14 minutes 45 seconds, the last index on the bias vector is n, when its supposed to in fact be a k. Thanks for the sharp eyes that caught that!
For those who want to learn more, I highly recommend the book by Michael Nielsen introducing neural networks and deep learning: goo.gl/Zmczdy
There are two neat things about this book. First, its available for free, so consider joining me in making a donation Nielsens way if you get something out of it. And second, its centered around walking through some code and data which you can download yourself, and which covers the same example that I introduce in this video. Yay for active learning! github.com/mnielsen/neural-networks-and-deep-learning
For those of you looking to go *even* deeper, check out the text «Deep Learning» by Goodfellow, Bengio, and Courville.
Also, the publication Distill is just utterly beautiful: distill.pub/
Lion photo by Kevin Pluck
— Timeline:
0:00 — Introduction example
1:07 — Series preview
2:42 — What are neurons?
3:35 — Introducing layers
5:31 — Why layers?
8:38 — Edge detection example
11:34 — Counting weights and biases
12:30 — How learning relates
13:26 — Notation and linear algebra
15:17 — Recap
16:27 — Some final words
17:03 — ReLU vs Sigmoid
— Animations largely made using manim, a scrappy open source python library. github.com/3b1b/manim
If you want to check it out, I feel compelled to warn you that its not the most well-documented tool, and has many other quirks you might expect in a library someone wrote with only their own use in mind.
If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then «add subtitles/cc». I really appreciate those who do this, as it helps make the lessons accessible to more people.
— 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted on new videos, subscribe, and click the bell to receive notifications (if youre into that).
If you are new to this channel and want to see more, a good place to start is this playlist: 3b1b.co/recommended