- [Jade] Thank you to Curiosity Stream
for sponsoring this episode.
Get access to my streaming video service, Nebula,
when you sign up for Curiosity Stream
using the link in the description.
(soft upbeat music)
- Hello, welcome to Up and Atom, I'm Jade.
What if I told you that there's a mathematical technique
where you can prove that you're right
without even telling anyone,
what it is that you're right about?
Sounds impossible, don't you think?
How can I possibly prove to you
that a secret I know is true without telling you the secret.
Like this,
we have a jar of candy clouds.
I reach in and choose two.
I swear that they're different colors
but I don't want you to know what the colors are.
How can I prove that they're different colors
without showing them to you
or telling you what the colors are?
Feel free to pause the video
and think about this if you'd like.
All right, I put the candy clouds under two cups.
When I close my eyes,
you either switch the places of the cups or leave them.
Then I look at what's under the cups and tell you correctly
whether you switched them after I guess correctly once,
you aren't going to be very convinced, right?
My chances of guessing
whether you switched them were 50-50 or 1 and 2.
So we do it again.
Because this statement I'm proving really is true.
The candy clouds are different colors.
I'm right again.
Now the chances of this being a fluke are one and four.
We can keep going as long as you want.
If you want to have a less than 1% chance
that I'm lying to you,
we'd only have to repeat this seven times.
If we repeated the swap
or don't swap routine 100 times a chances
of me answering correctly every time
without knowing the candy clouds with different colors
is less than 7.89 times 10 to the power of minus 31,
practically impossible.
This number, isn't actually zero.
So I haven't given a mathematical proof
that I can't be lying, but if aren't satisfied
we can keep going until the odds that I'm lying
are smaller than any non-zero number you choose.
Then we say that I have proven that I know a true secret
and you have verified that I know a true secret.
We didn't just find this on a piece of paper somewhere.
We interacted to construct this proof.
Our candy cloud game satisfies all the requirements
of what's called a zero knowledge proof.
A special type of interactive proof
in which one party demonstrates they know something
without giving away any information.
In our example, I was the prover and you were the verifier.
I proved that I had the secret
by answering all the questions correctly.
And you verified it by knowing
whether you had switched the cups
and therefore knowing what the answer should have been
swapped or not.
Now, let's think about how we did this.
To know that we'd be able to construct
a zero knowledge proof.
We'd both had to be honest.
I was telling the truth.
I really did know the secret
that the candy clouds were different colors.
We both followed the protocol.
You told the truth about
whether you had switched the cups each time.
And I correctly told you whether or not you did too.
This property of a zero knowledge proof
is called completeness.
Whenever the prover really has a secret
and both parties follow the protocol
the verifier will agree
that the prover really does know the secret.
That covers the idea that we want to be able
to prove things that are true.
Soundness is what guarantees
that we can't prove anything false.
In practice this means we can get the chance
that I'm lying to be as small as we want.
I wouldn't have been able to correctly say
whether you'd switched the cups each time,
unless my claim that the candy clouds
were different colors was true.
And finally, you have no more knowledge at the end
than you had at the beginning.
You don't know the colors of the candy clouds
or which one is which you only verified what I told you
that they were different colors.
This is called zero knowledge.
Not only can the verifier not cheat
and learn more than what they're supposed to
but an eavesdropper hearing swapped.
Didn't didn't, didn't swapped, swapped didn't,
and so on.
Wouldn't be able to tell whether
we made the whole up
or whether we'd really verified something
because only you know, whether you switched the cups.
Therefore only we know that my answers were correct.
So to qualify,
a zero knowledge proof needs to satisfy
all three of those conditions, completeness, soundness
and zero knowledge.
If you find the existence of zero, your knowledge proofs
hard to believe you're not alone.
The early reviewers of the groundbreaking paper
must not have believed they were possible either
because the paper was rejected multiple times
when it finally was accepted its authors,
Shafi Goldwasser and Silvio Micali
were awarded the A.M.Turing award
by the association for the computing machinery,
generally considered the highest award
in computer science sometimes called
The Nobel prize of computer science.
Okay, so this is pretty cool,
but can we actually use it for anything?
Just six or seven years ago?
There were no "real world" applications
of zero knowledge proofs
but that has changed with rising popularity
of cryptocurrency.
For the very first cryptocurrency, Bitcoin
the inventor essentially had to make a choice
between trust and privacy.
In order for people to believe they weren't being swindled.
There needed to be a permanent unalterable public record
of every Bitcoin transaction.
This was called the blockchain.
The issue is
if that one person used Bitcoin
to pay for something like cancer treatment,
that transaction is public.
Anyone can now see that user Jane Bit
paid for cancer treatment.
That's a pretty big compromise of privacy.
Enter zero knowledge proofs.
For some cryptocurrency
people don't have to make such public transactions anymore.
Zero knowledge proof algorithms have advanced to the point
that they can be used to verify the blockchain
without actually saying what's in it.
The contents of the record of every transaction
the blockchain is a secret
and zero knowledge proofs are being used to verify
that the record is correct
without revealing any information.
Another cool thing about zero knowledge proofs
is that every mathematical proposition
that has a proof also has a zero knowledge proof.
Okay, so a proposition is just a mathematical statement
that is either true or false.
For example, the sum of angles
in a triangle is 180 degrees is a true proposition
in nuclearity and geometry.
The interior angles of a square add up to 300 degrees
is a false proposition in nuclearity and geometry.
Now let's consider what is perhaps
the most famous open question in all of mathematics,
the Riemann hypothesis.
The statement of the hypothesis is a proposition.
We just don't know if it's true.
It turns out that if you have a proof
of the Riemann hypothesis
then you can prove to me that you really do have a proof
without giving away any information about the proof.
In other words, if there's a proof of the Riemann hypothesis
there's a zero knowledge proof of the Riemann hypothesis.
And this applies to all mathematical propositions
with the proof.
Isn't that wild what's even crazier
is that this result was proven by coloring in graphs.
Let me know in the comments,
if you'd like me to make a follow up video
about how it was done.
Zero knowledge proofs, change the idea of a proof
from a static object that anyone could pick up and read
to an interactive game between two players.
This is a fundamental advancement
to computer science and cryptography.
I have a confession to make
for someone who spends their days,
reading about computer science,
I know embarrassingly little about crypto currency.
Studying for this video was the first time I learned
what the blockchain actually was.
Anyway, I got pretty curious
and wanted to get with the times.
So I found a really nice documentary
on Curiosity Stream called "The Blockchain Revolution".
I learned what the blockchain was
how it could impact not only the future of money
but autonomous vehicles,
renewable energy and digital identity.
It's a revolutionary idea, much bigger than just money.
But I also learned about its downsized, like,
how it caused people to lose millions of dollars in a scam.
If you're interested in learning more about the blockchain
and how it works,
you can watch this documentary for free
by signing up to Curiosity Stream
with the link in the description.
Curiosity Stream is an award-winning documentary service
with thousands of titles from space exploration
to the birth of the internet,
to cyber wars and nanotechnology.
They're also supporting a bunch of us,
educational YouTube creators
with our own streaming platform, Nebula,
where Curiosity Stream
is all about big budget documentaries.
We are building Nebula because we want a place
for education E-creators to try our new content ideas
that might not work on YouTube.
I've made a documentary about whether math is invented
or discovered, which you can check out if you want.
Curiosity Stream loves independent creators
and wants to help us grow our platform.
So they're offering up and atom viewers,
free access to Nebula,
when you sign up at curiositystream.com/upandatom.
They're also offering a 26% discount to up and atom viewers.
So that's two streaming services for just $14.79 for a year.
By signing up to Curiosity Stream,
you'll be helping not just me,
but the entire educational community.
Thank you for watching
and thank you to all my Patron supporters.
I'll see you in the next episode.
Bye.
(soft upbeat music)