Cryptography 101 for Blockchain Developers Part 3/3: Elliptic Curve Pairings

Cryptography Fundamentals for blockchain developers. The third and final video in a series to learn the building blocks of cryptography ultimately setting up the foundation for Zero-knowledge proofs. In this video, KoalateeCtrl (Blockchain Security Researcher @ OpenZeppelin) will walk you through the basics of Elliptic Curve Pairings specifically for developers.

An introduction to elliptic curve pairings for blockchain developers and folks in Web3 that want to dive deeper under the hood and better understand cryptography. In addition, if you are interested in cryptographic structures such as elliptic curve pairings, zero knowledge proofs, signatures schemes, and their applications, make sure to first watch this video to build a solid foundation.

This video is part 3 of a 3 part series that should set you up to begin to pursue more advanced cryptography.

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TIMESTAMPS:
0:00 Introduction
1:40 Elliptic Curve Group Review
3:01 Motivation for Pairings
10:01 Elliptic Curve Pairings
17:15 Pairings for Verification
21:42 Recap

#cryptography #pairings #ellipticcurve #blockchain #web3 #developers #grouptheory

[sweeterasmus]

Will you please continue with these series?

[priyankgupta]

My brain is officially fried learning about al of this. How the hell did you manage to put this together? I could never do that. Kudos to you dude.

[jinwooseong]

The myth of Elliptic curve is finally gone in my brain, thank you so much for this greeeeaaaat content!

[gleleylo]

A very concise and clear video, thank you. It's very interesting to know how e(A, B) is implemented.

[Aramik-lpfn]

This is one of the best tutorial videos about pairings. I always had the same question of why the pairings were invented in the first place and what was the benefits . This answered all my questions in that regard. Thanks a lot!

[jonasp]

Well I need an implementation of the function e for this to work. The logic is clear, but a prerequisite to implementing it is an understanding of a real application of e.

[robfielding]

i see something that may be a connection to Clifford Algebra (ie: Geometric Algebra). Imagine that G1 is a vector, and G2 is an orthogonal vector. In that case: (G1 G2) would be a rotation. If all the groups are written additively:

e( a G1, b G2) = a b (G1 G2)

in Geometric Algebra, e1 and e2 are orthogonzl directions in space. And (e1 e2) squares to -1. i can't shake this similarity. if we had a way to represent a scalar group as just "1", then you could think of e() as the geometric product.

a e1 * b e2 = a b (e1 e2)

(a + c)e1 * b e2 = (a + c)b (e1 e2)

In Geometric Algebra, they have taken the number system, and added directions in space as primitives.

Note that when you multiply two vectors that are orthogonal e1, e2, you fall into the complex numbers which is a sum of multiples of 1, (e1 e2).

[liffidmonky]

1:47 You should have said we are all in the same group

[quonxinquonyi]

Isn’t elliptic curve pairing isogeny ???