Elliptic Curve Cryptography Overview 

how is this guy so good at writing backwards...

2:20 wrong; the product of two prime numbers is always non-prime - because it has the two prime numbers as factors.

I never knew Matthew McConaughey was so good at math

found this while prepping for the interview. thank you for such simple and yet practical explanation!

Love how you have to plug the BIG-IP thing at the end (someone has to pay for the Light Board. Well done, sir. You are a great presenter. One of the best I've come across.

Protip: doing G + G is the equivalent of finding the point tangent to the curve at G! And since we already have added "two" points (the curve doesn't care if the points are different), the curve will only intersect at one other point!

Good introduction to ECC. In you intro to RSA you mention taking random prime numbers and multiplying them to get a really big prime number. The result is a really big composite number(not prime) that is hard to factorize.

this is by far the best video i have come across. Simple, explained in layman's terms to beginners and under 15 minutes. rate this 10 out of 10

I have read about this before, but this is clearly explained! Well done.

This video broke it down so well!! Thank you!!

This is a great easy-to-understand intro to ECC!

Fantastic work John!

Great video. I have a question, if you have the power potencial to multiply your private key and the "generator point" to get your public key, can you get the private key if you have the public key and the "generator point"? I mean iterating over and over and saving every result until match with your public key (that can be the same process that you used to get it at the first time). Thanks

I have always operated under the knowledge that multiplying two prime numbers will result in a number that is certainly not prime, as it will have factors 1, itself, and the two numbers used to generate it?

The best explanation that I got on Elliptic Curve Cryptography , great work John

This was a fantastic intro to ECC, thanks for the clear explanation!

Excellent presentation. The Elliptic Curve in the video is drawn based on y^2=x^3-3x+5. The actual elliptic curve used in the algorithm will have much bigger prime numbers and will look much different. The same logic applies to either case, so it doesn't quite matter. Just for your information.

Thanks, brother, you're a huge help in my Crypto class!

I almost got an idea wats was it all about...thank you John Wagnon

A very clear and interesting explanation! Thanks!

Thank you for this wonderful video !

Thank you! Beautifully explained.

This video is, by far, the best video on eliptic curve criptography availiable... wish you could do more videos about this subject, congratulations for the amazing work!!!

Thanks for your wonderful presentation.

And what exactly is then the “dot” operation algorithmically? A . B means find another point on the intersection of the line connecting A and B with the elliptic curve?

Fantastic work on the video! A lot of smart people forget that it is hard to learn things when they make it super complicated; I hope that I can be as good as you one day :D. I thought I would summarize the video for myself (and others if they might also benefit from it?) and ask a few questions. From what I understand, elliptic curve cryptography uses fewer bits to create as complex of a trapdoor function as RSA (which is basically trying to factor a really large semi-prime number). In elliptic curve cryptography, you start with two points on this elliptic curve (looks like an octopus and is symmetric about y-axis) and you find the third point you find when you draw a line between those to find another point on the graph and then find the point symmetric to that about the x-axis. E.g. If the initial two points were A and B, the third point would be notated as A + B = C. Then you do A + C to find D. And then you do A * D = E … and so on until you find some point Z on this elliptic curve. The number of these additions you have to do to acquire Z is the “private key”, which is why this computation is often written as K = k * G, where k is the private key, K is the point we are trying to reach, and G is the generator point (the point we start at and is constant for a certain graph). Some questions I have are:
 1) From what I understand, exponentiation by squaring makes this logarithmically easier and allows one to verify this quickly - but how does one square these “dot products”? These are not just vectors going through some kind of constant transformation when you do A * B (at least from what I understood). I have now found that addition is commutative in elliptic curves! So doing 4 * G can be simplified as (G + G) + (G + G); or basically that you can break down a multiplication in about log_2(k) steps. 2. How do you encode a point on a graph as one number? Would you just encode the y values? I am not sure that it would be super helpful towards finding the squares of two numbers though. I found out that that the first half of the bits are the x coordinates and the second half are the y coordinates. 3. What is the reason behind the reduction in bits of elliptic curve cryptography? Still not sure about this! 4. What exactly happens when a point goes over the maximum? You find how much it is above the maximum, and make it that much more than the x minimum (how do you decide if the new value is positive or negative)? What happens if the new x value is also more than the maximum? Do you just keep on moving the value back until the value is below the maximum?

Thanks for providing this really useful intuition on the algorithm.

It took me a while to realize and appreciate that this dude is writing backwards so we can read it forwards. Also, love your eyeballs. They are grade A, top-shelf eyeballs.

That was one very clear explanation of ECC. How can there be any thumbs down AT ALL?

Thank you! You have satisficed my curiosity

Above all, thanks for keeping the vid titles short

this is extremely well explained, thanks

is there any relation between discrete logarithm and integer factorization?

Amazingly efficient explanation. How does this channel have so few viewers?

i have problems understanding from the point where he draws the curve... i wonder if there are better resources on ecc?

Fantastic overview! Thank you!

I'm impressed by writing the mirror images of text on the screen but this doesn't actually explain how anything is calculated. The dots are an operator which represents the number of intersections? How does the straight line re-intersect with the curve after it wraps around?

You have saved my math project. Thank you

Great explanation of how EC works, but how does someone actually use this to encrypt something, say like a message or a string? Would the receiving party need the same private key to decrypt?

Well and simply explained, good job!

I could not get this concept at all until I watched you're video. Thank you very much.

Thanks for your wonderful presentation

This video is really interesting! Keep up the good work.

Very good introduction. I do get the basic idea of it now.

Man i love your videos,Thank you so much

And I thought ECC stood for ERP Central Component :). Thanks for the great presentation!

Hello sir How can I encoded the message to a point in ECC over the prime field?

just a technicality: eliptic curve can't be defined as a function, it's more of a formula. That's why it's called a curve, not a math function.

You can merge RSA and ECC to become RSA more effective?

Thanks for the nice intuition!

this recursion is glossed over often, thanks

This video was super helpful thanks so much

I'm sorry i don't understand how can u deal with values which are higher than the max value ? Thanks

But if we keep gradually increasing the number of time of "dot" function and try every output as it goes, doesn't it mean we could eventually reach the right Z? My impression is the only thing Z depends on is just the number of "dot" function, while I'm aware that factoring a number into primes in RSA is hard since it's a combinatorial problem. Am I missing something?

It baffles me how people have the knowledge enough to 1) Come up with such ideas and most importantly 2) To code such applications that can do such complex things. The cryptographic world is so unique in so many ways, as us people many times take it for granted to ease of use in such applications since we can freely use them, but lord knows the backend behind all that computation

very clear explanation. thanks you sir.

Great video! You explained this better than my uni prof XD

You're best one-way-function-teacher I have ever met. Finally I find out what the point and sense is. (I saw this super-interactive presentation technique but never discovered what do you use for the transparency and written-in-air effect. I know that somewhere in the RU-vid is it descripted but I can'find it... Can you divulge it to me? :)

Fun fact, but my integral Calculus teacher in university was one of the creators of this :) Neil Koblitz. Very smart dude.

Excellent overview.

great explanantion sir,for such complicated topic ..i hv exam tomorow and this gonna help me loooottttttt thnxaaaa tunssss.god bless u

i'm very impressed about your skills writing backward

Fantastic, clear and well made video. Got here while reading “Mastering Ethereum” and wanting a more thorough understanding of ECC maths. Got what I wanted!

Great Video! Going to write a paper about ECC, this helped a lot.

thank you so much, you explained it so well

Great video-At 2.24, you mentioned multiply 2 big prime numbers together and you get a larger prime number? I think you meant you get a large composite number.

There are a few confusing remarks such as "dotting A with itself" and "dotting the curve with itself", neither of which appears to have any clear meaning. The latter was probably just a slip of the tongue in trying to finish up in a hurry, but the other is repeated a few times. If we start with the curve and point A how can we possibly know where dotting A with itself will place B? Don't we need two points to get started? Or is the dot operation given by some formula that wasn't specified? That wouldn't seem to make much sense either because then the whole path A -> Z is completely determined by just the curve and A.

Perfect. Thanks and great job.

that was a really, really good intro to this topic

Awesome video!

Great explanation.

How do you measure the equivalence in length of keys of both algorithms?

how did the sender created the encrypted message without knowing "n"?how many times did he run a through the function? the sender and the receiver must meet to share the value of n so this isn't PGP encryption

Very informative. Thank you.

This was really well explained

Great informative video!

This is very good video to eplain ECC. Thanks

very outstanding explanation sir

very complex topic ... how do you encryp or decrypt messages then ? even though you only SIGN messages, most of the times ?

This is awesome!

Excellent video, thanks a lot

Hold on though... on the Diffie Hellmann algorithm integers are chosen randomly. In the ECC algorithm you have to calculate the nth term by using a sequence of dot operations to arrive at your private number. The point is, calculating your private key is a linear sequence of prescribed operations with a finite terminating point. All an attacker would have to do is run through the operations themselves, which they could do.... they dont know where you stopped but they only need to find the first one that works. And if you can computer yours, surely they can compute it too.

What would be SUPER useful in my opinion is show that if G is the generator point demonstrate that 1G + 2G + 8G = 11G - becuase it's the commutative property that MUST be used to do multiplication, right? If it is, I think people will see why point multiplication is so easy, but point division is basically so difficult. In the example I gave, 11 would be the secret key, wouldn't it? You know the final point, but you don't know what the original number was for scalar multiplication. This was my difficulty in understanding this.

This is the best explanation!

But how do you actually encrypt something with that private key?

Interesting topic! Learned a lot. You type backwards really well. :-)

Nice Explanation!!

This was incredibly helpful. Thank you!

thanks professor ... well done

Man these videos are so good, even my dumbass is able to understand these topics, thanks John!

Do you use RSA to lock down that n?

Nice short explanation of a very complex subject. Actually I am looking at such explanations to help me to understand whether relatively simple 'dotting' is commonly used in implementations or whether 'point doubling' (where the line is a first derivative 'tangent' to the curve rather than a sort of chord) is actually used. IMO you got the distinction wrong there with regard to 'dotting with itself' which is 'point doubling' not exactly the same as 'dotting'. Also, the reflecting across the x axis is, I believe, part of the group operation. That is to say that A dot B gives you an intermediate point which when reflected gives you C. Anyway, good job with the video. It is difficult to simplify without losing some perhaps trivial distinctions, and you probably know more about it than I do in the long run.

What kind math do need to learn cryptographic?just wondering

Brilliant Introduction

Can you please provide c#code to encrypt / decrypt string using ECC

Great video

Great explanation!

A good way to get a general idea of ECC

Good video. Thanks

great, good content