About
- ZKDL Camp is a series of internal lectures on Zero-Knowledge Proving Systems conducted internally at Distributed Lab. Here, we will understand "from scratch" how SNARKs (Groth16, PLONK), STARK, Bulletproofs (in the future, perhaps also Folding schemes) work. + ZKDL Camp is a series of internal lectures on Zero-Knowledge Proving Systems at Distributed Lab. Here, we will understand "from scratch" how SNARKs (Groth16, PLONK), STARK, Bulletproofs (in the future, perhaps also Folding schemes) work.
It is important that this course is designed for a full low-level understanding of these protocols and, accordingly, all the mathematics on which they are based. That is why the course covers not only zk itself and its application directly but also the basic level of mathematics needed to understand zk and cryptography in general. @@ -60,25 +60,27 @@
About
Lectures
- +
+ 
Lecture 1: Mathematics for Cryptography I
+Lecture 4: Projective Coordinates and Pairing
Speaker: Dmytro Zakharov
- Content. In this lecture, we will cover the basic mathematics required for understanding Zero-Knowledge Proving Systems. Here, we cover: -
-
-
- Basic Set Theory and Logic -
- Basic Group Theory -
- Fields. Finite Field Arithmetic. + Content. This lecture will touch the central part of SNARKs: elliptic curve pairing, its properties and applications. But first, we will cover projective coordinates and how they can be used to optimize elliptic curve operations. All in all, here we cover: +
- Relations and equivalence classes. +
- Projective Coordinates. +
- Adding points in projective coordinates. Scalar multiplication. +
- Elliptic Curve Pairing. +
- Pairing applications.
-
+
Lecture 1: Mathematics for Cryptography I
Lecture Material:
+ +Other Helpful Resources:
+-
+
- "Pairing for Beginners" by Craig Costello: Not only this already mentioned book contains pairing inner working, but also a profound introduction into projective form of elliptic curves. +
- "A Graduate Course in Applied Cryptography" by Dan Boneh and Victor Shoup, Section 15 Recommended to read this chapter up to section 15.5 including. +
- "BN254 For The Rest of Us" by Jonathan Wang: if you want specifics of pairing-friendly curve parameters, you can take a look here. This curve is currently widely used. +
- High-Speed Software Implementation of the Optimal Ate Pairing over Barreto-Naehrig Curves by Jean-Luc Beuchat et al.: if you really want to get into the details of pairing implementation, this paper is the best. It contains all the algorithms to implement pairing from scratch. +
+ Lecture 3: Finite Field Extensions and Elliptic Curves
+Speaker: Dmytro Zakharov
++ Content. Our primary focus will be on Finite Field Extensions, while also covering basics of Elliptic Curves. Here, we cover: +
-
+
- Finite Field Extensions +
- Algebraic Closure +
- Elliptic Curve Definition +
- Discere Logarithm on Elliptic Curves +
Lecture Material:
+ +
+ Lecture Notes
+ + +
Slides
- +
Exercises
- +
Test Form
@@ -112,9 +175,10 @@Lecture Material:
Other Helpful Resources:
-
-
- Michael Penn's "Abstract Algebra" Youtube Course : one of the best explanations of Algebra I have seen with tons of examples. -
- MIT Modern Algebra Class (18.703) : Quite a rigorous course in Algebra, yet still easy-to-follow. -
- Visual Group Theory : Very comprehensive list of lectures on Group Theory and Galois Theory with illustrations and many examples. +
- An Introduction to the Theory of Field Extensions by Samuel Moy: rigorous and "mathematicalish" introduction to what field extensions are. Highly recommended, but maybe too much for basic understanding of finite field extensions. +
- "Guide to Elliptic Curve Cryptography" by Alfred Menezes et al.: Amazing book on Elliptic Curves optimizations. It also contains a great introduction into finite fields. +
- "Pairing for Beginners" by Craig Costello: Quick dive into inner workings of elliptic curve pairing. +
- "A Graduate Course in Applied Cryptography" by Dan Boneh and Victor Shoup, Section 15 (Elliptic curve cryptography and pairings): Elliptic Curves are described very well together with the basics of pairing (without going into the details of construction).
Other Helpful Resources:
+
Lecture 3: Finite Field Extensions and Elliptic Curves
+Lecture 1: Mathematics for Cryptography I
Speaker: Dmytro Zakharov
- Content. Our primary focus will be on Finite Field Extensions, while also covering basics of Elliptic Curves. Here, we cover: -
-
-
- Finite Field Extensions -
- Algebraic Closure -
- Elliptic Curve Definition -
- Discere Logarithm on Elliptic Curves + Content. In this lecture, we will cover the basic mathematics required for understanding Zero-Knowledge Proving Systems. Here, we cover: +
- Basic Set Theory and Logic +
- Basic Group Theory +
- Fields. Finite Field Arithmetic.
-
+
Lecture 3: Finite Field Extensions and Elliptic
Lecture Material:
-
+
Lecture Notes
-
+
-
+
Exercises
-
+
Test Form
@@ -232,10 +295,9 @@ Lecture Material:
Other Helpful Resources:
- - An Introduction to the Theory of Field Extensions by Samuel Moy: rigorous and "mathematicalish" introduction to what field extensions are. Highly recommended, but maybe too much for basic understanding of finite field extensions.
- - "Guide to Elliptic Curve Cryptography" by Alfred Menezes et al.: Amazing book on Elliptic Curves optimizations. It also contains a great introduction into finite fields.
- - "Pairing for Beginners" by Craig Costello: Quick dive into inner workings of elliptic curve pairing.
- - "A Graduate Course in Applied Cryptography" by Dan Boneh and Victor Shoup, Section 15 (Elliptic curve cryptography and pairings): Elliptic Curves are described very well together with the basics of pairing (without going into the details of construction).
+ - Michael Penn's "Abstract Algebra" Youtube Course : one of the best explanations of Algebra I have seen with tons of examples.
+ - MIT Modern Algebra Class (18.703) : Quite a rigorous course in Algebra, yet still easy-to-follow.
+ - Visual Group Theory : Very comprehensive list of lectures on Group Theory and Galois Theory with illustrations and many examples.
Lecture Notes
- + - +
Exercises
- +
Test Form
@@ -232,10 +295,9 @@Lecture Material:
Other Helpful Resources:
-
-
- An Introduction to the Theory of Field Extensions by Samuel Moy: rigorous and "mathematicalish" introduction to what field extensions are. Highly recommended, but maybe too much for basic understanding of finite field extensions. -
- "Guide to Elliptic Curve Cryptography" by Alfred Menezes et al.: Amazing book on Elliptic Curves optimizations. It also contains a great introduction into finite fields. -
- "Pairing for Beginners" by Craig Costello: Quick dive into inner workings of elliptic curve pairing. -
- "A Graduate Course in Applied Cryptography" by Dan Boneh and Victor Shoup, Section 15 (Elliptic curve cryptography and pairings): Elliptic Curves are described very well together with the basics of pairing (without going into the details of construction). +
- Michael Penn's "Abstract Algebra" Youtube Course : one of the best explanations of Algebra I have seen with tons of examples. +
- MIT Modern Algebra Class (18.703) : Quite a rigorous course in Algebra, yet still easy-to-follow. +
- Visual Group Theory : Very comprehensive list of lectures on Group Theory and Galois Theory with illustrations and many examples.