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<!DOCTYPE html>
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<html lang="en">
<head>
<title>ZKDL Camp</title>
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<meta charset="UTF-8">
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<meta property="og:url" content="https://zkdl-camp.github.io/" />
<meta property="og:type" content="website">
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<link rel="stylesheet" href="styles/main/main.css">
</head>
<body>
<!-- ZKDL Header -->
<div class="title-block">
<h1 class="title-text"> ZKDL <br/> Camp </h1>
</div>
<!-- About Section -->
<div class="about-block">
<h1 class="section-header"> About </h1>
<p>
<b> ZKDL Camp </b> is a series of lectures on zero-knowledge (ZK) proofs, conducted at <a href="https://distributedlab.com/">Distributed Lab</a>, in which we explore "from scratch" how modern <i>zk-SNARKs</i> such as <i>Groth16</i>, <i>PlonK</i>, <i>GKR</i> work, including all the mathematical components they rely on.
</p>
<p>
Note 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 theory itself and its applications, but also the basic mathematics level needed to understand ZK and cryptography in general.
</p>
</div>
<!-- About Section -->
<div class="summary-block">
<h1 class="section-header"> Book </h1>
<p>
Based on the lecture material, we composed a book, which is
available by the link below. Note that the book is still in
active development, and we are looking forward to your feedback
on it. If you want to contribute, visit our <a
href="https://github.com/ZKDL-Camp/lectures-tex">GitHub
Repository</a> and leave your comments or pull requests.
</p>
<div class="book-container">
<img src="images/book_cover.png" alt="Book Cover" class="book-cover">
<div class="button-container"></div>
<a href="files/summary/lecture-notes.pdf" class="open-book-button" target="_blank">Open Book</a>
</div>
</div>
</div>
<!-- Lectures Section -->
<div class="lectures-block">
<h1 class="section-header"> Lectures </h1>
<!-- Lecture 18 -->
<div class="lecture-card">
<div class="lecture-card-head">
<div class="lecture-card-preview">
<img src="images/lecture-previews/lecture_18.png" class="preview">
</div>
<div class="lecture-card-content">
<div class="lecture-card-header">
<h2 class="lecture-card-title"> Lecture 18: UltraGroth </h2>
<p class="lecture-card-speaker"> <strong>Speaker:</strong> Dmytro Zakharov </p>
</div>
<div class="lecture-card-content">
<p class="lecture-card-description">
<strong>Content.</strong> Embedding lookup checks
into Groth16 is unfortunately currently impossible.
UltraGroth is a protocol designed to insignificantly
modify Groth16 to introduce lookup checks that
optimize arithmetical circuits written over R1CS. In
this lecture, we consider the construction of UltraGroth,
how to build interactive protocol over multi-round QAP, and efficiency
of such construction.
</p>
</div>
</div>
</div>
<div class="lecture-links-section">
<h1 class="lecture-links-header"> <strong> Lecture Material: </strong> </h1>
<div class="lecture-links-list">
<a href="https://hackmd.io/@ZamDimon/SkFGEXbKgx" class="lecture-link">
<img src="images/icons/hackmd.png">
<p class="link-name"> Lecture Notes </p>
</a>
<a href="files/slides/18-ultragroth-informal.pdf" class="lecture-link">
<img src="images/icons/slides_icon.png">
<p class="link-name"> High-level Overview </p>
</a>
<a href="files/slides/18-ultragroth-formal.pdf" class="lecture-link">
<img src="images/icons/slides_icon.png">
<p class="link-name"> Technical Slides </p>
</a>
</div>
<h1 class="lecture-links-header"> <strong> Other Helpful Resources: </strong> </h1>
<ul>
<li> <a href="https://hackmd.io/@Merlin404/Hy_O2Gi-h"> "UltraGroth" </a> by Lev Soukhanov: original blog on UltraGroth construction. </li>
<li> <a href="https://mirror.xyz/0x90699B5A52BccbdFe73d5c9F3d039a33fb2D1AF6/T4lHiBlo7VoYp-5SEuitQnY7_ullpwkZbanOOwjeZI4"> "Implementing Lookups in Groth: The second ingredient to reduce Bionetta’s constraints" </a> by Illia Dovhopolyi, Artem Sdobnov, Dmytro Zakharov: application of UltraGroth to zero-knowledge Machine Learning (zkML) problem. </li>
</ul>
</div>
</div>
<!-- Lecture 17 -->
<div class="lecture-card">
<div class="lecture-card-head">
<div class="lecture-card-preview">
<img src="images/lecture-previews/lecture_17.png" class="preview">
</div>
<div class="lecture-card-content">
<div class="lecture-card-header">
<h2 class="lecture-card-title"> Lecture 17: Lookup Tables </h2>
<p class="lecture-card-speaker"> <strong>Speaker:</strong> Dmytro Zakharov </p>
</div>
<div class="lecture-card-content">
<p class="lecture-card-description">
<strong>Content.</strong> This lecture introduces
a powerful tool to reduce the number of constraints
in the arithmetical circuits representing complex
non-native logic in the given zero-knowledge protocol. We
consider two widely used lookup protocols: plookup and
logup.
</p>
</div>
</div>
</div>
<div class="lecture-links-section">
<h1 class="lecture-links-header"> <strong> Lecture Material: </strong> </h1>
<div class="lecture-links-list">
<a href="files/notes/17-lookups.pdf" class="lecture-link">
<img src="images/icons/lecture_notes_icon.png">
<p class="link-name"> Lecture Notes </p>
</a>
<a href="files/slides/17-lookups.pdf" class="lecture-link">
<img src="images/icons/slides_icon.png">
<p class="link-name"> Slides </p>
</a>
</div>
<h1 class="lecture-links-header"> <strong> Other Helpful Resources: </strong> </h1>
<ul>
<li> <a href="https://eprint.iacr.org/2020/315"> "plookup: A simplified polynomial protocol for lookup tables" </a> by Ariel Gabizon and Zachary J. Williamson: Original plookup paper. </li>
<li> <a href="https://hackmd.io/@relgabizon/ByFgSDA7D"> "Multiset checks in PLONK and Plookup" </a> by Ariel Gabizon: simplified explanation of plookup protocol from its author. </li>
<li> <a href="https://eprint.iacr.org/2022/1530.pdf"> "Multivariate lookups based on logarithmic derivatives" </a> by Ulrich Habock: Original Logup paper. </li>
</ul>
</div>
</div>
<!-- Lecture 16 -->
<div class="lecture-card">
<div class="lecture-card-head">
<div class="lecture-card-preview">
<img src="images/lecture-previews/lecture_16.png" class="preview">
</div>
<div class="lecture-card-content">
<div class="lecture-card-header">
<h2 class="lecture-card-title"> Lecture 16: GKR and Offline Memory Checking </h2>
<p class="lecture-card-speaker"> <strong>Speakers:</strong> Dmytro Zakharov and Anton Levochko </p>
</div>
<div class="lecture-card-content">
<p class="lecture-card-description">
<strong>Content.</strong> This lecture introduces
the first application of Sum-Check protocol for
building SNARK over Circuit Satisfiability problem.
Additionally, we specify how Sum-Check is
used for Offline Memory Checking problem.
</p>
</div>
</div>
</div>
<div class="lecture-links-section">
<h1 class="lecture-links-header"> <strong> Lecture Material: </strong> </h1>
<div class="lecture-links-list">
<a href="files/notes/16-gkr.pdf" class="lecture-link">
<img src="images/icons/lecture_notes_icon.png">
<p class="link-name"> Lecture Notes </p>
</a>
<a href="files/slides/16-gkr.pdf" class="lecture-link">
<img src="images/icons/slides_icon.png">
<p class="link-name"> Slides </p>
</a>
</div>
<h1 class="lecture-links-header"> <strong> Other Helpful Resources: </strong> </h1>
<ul>
<li> <a href="https://www.microsoft.com/en-us/research/wp-content/uploads/2016/12/2008-DelegatingComputation.pdf"> "Delegating Computation: Interactive Proofs for Muggles" </a> by Shafi Goldwasser, Yael Tauman Kalai, and Guy N. Rothblum: original GKR paper. </li>
<li> <a href="https://people.cs.georgetown.edu/jthaler/GKRNote.pdf"> "A Note on the GKR Protocol" </a> by Justin Thaler: Simplification of original GKR interactive proof for circuit evaluation. </li>
</ul>
</div>
</div>
<!-- Lecture 15 -->
<div class="lecture-card">
<div class="lecture-card-head">
<div class="lecture-card-preview">
<img src="images/lecture-previews/lecture_15.png" class="preview">
</div>
<div class="lecture-card-content">
<div class="lecture-card-header">
<h2 class="lecture-card-title"> Lecture 15: Sum-Check Protocol </h2>
<p class="lecture-card-speaker"> <strong>Speaker:</strong> Dmytro Zakharov </p>
</div>
<div class="lecture-card-content">
<p class="lecture-card-description">
<strong>Content.</strong> This lecture introduces
Sum-Check protocol which is a cornerstone of many modern
zk-SNARKs. Additionally, we introduce the notion of
multilinear polynomials and how to encode certain problems
into the multilinear form.
</p>
</div>
</div>
</div>
<div class="lecture-links-section">
<h1 class="lecture-links-header"> <strong> Lecture Material: </strong> </h1>
<div class="lecture-links-list">
<a href="files/notes/15-sum-check.pdf" class="lecture-link">
<img src="images/icons/lecture_notes_icon.png">
<p class="link-name"> Lecture Notes </p>
</a>
<a href="files/slides/15-sumcheck.pdf" class="lecture-link">
<img src="images/icons/slides_icon.png">
<p class="link-name"> Slides </p>
</a>
</div>
<h1 class="lecture-links-header"> <strong> Other Helpful Resources: </strong> </h1>
<ul>
<li> <a href="https://people.cs.georgetown.edu/jthaler/ProofsArgsAndZK.html"> "Proofs, Arguments, and Zero-Knowledge" </a> by Justin Thaler: Book with a comprehensive description of Sum-Check protocols, its modifications, and applications. </li>
<li> <a href="https://www.youtube.com/watch?v=4018OYyoAf8"> "ZKP MOOC Lecture 4: Interactive Proofs" </a> by Justin Thaler: Lecture on Interactive Protocols and, in particular, Sum-Check protocol. </li>
</ul>
</div>
</div>
<!-- Lecture 14 -->
<div class="lecture-card">
<div class="lecture-card-head">
<div class="lecture-card-preview">
<img src="images/lecture-previews/lecture_14.png" class="preview">
</div>
<div class="lecture-card-content">
<div class="lecture-card-header">
<h2 class="lecture-card-title"> Lecture 14: Bulletproofs </h2>
<p class="lecture-card-speaker"> <strong>Speaker:</strong> Yevhen Hrubian </p>
</div>
<div class="lecture-card-content">
<p class="lecture-card-description">
<strong>Content.</strong> In this lecture, we consider
Bulletproofs. This zero-knowledge proof system has logarithmic proof size,
requires no trusted setup, and requires minimal cryptographic
assumptions: only the discrete log assumption. In this lecture,
we introduce inner-product argument and corresponding
polynomial commitment scheme, and explain how to use it
for range proofs and arithmetic circuits satisfiability.
</p>
</div>
</div>
</div>
<div class="lecture-links-section">
<h1 class="lecture-links-header"> <strong> Lecture Material: </strong> </h1>
<div class="lecture-links-list">
<a href="files/notes/14-bulletproofs.pdf" class="lecture-link">
<img src="images/icons/lecture_notes_icon.png">
<p class="link-name"> Lecture Notes </p>
</a>
<a href="files/slides/14-bulletproofs-part-1.pdf" class="lecture-link">
<img src="images/icons/slides_icon.png">
<p class="link-name"> Slides (Part I) </p>
</a>
<a href="files/slides/14-bulletproofs-part-2.pdf" class="lecture-link">
<img src="images/icons/slides_icon.png">
<p class="link-name"> Slides (Part II) </p>
</a>
<!-- <a href="https://github.com/ZKDL-Camp/circom-walkthrough" class="lecture-link">
<img src="images/icons/github.png">
<p class="link-name"> GitHub Repo </p>
</a> -->
</div>
<h1 class="lecture-links-header"> <strong> Other Helpful Resources: </strong> </h1>
<ul>
<li> <a href="https://doc-internal.dalek.rs/bulletproofs/index.html"> "Crate bulletproofs" </a> by dalek-cryptography: A pure-Rust implementation of Bulletproofs using Ristretto. </li>
<li> <a href="https://eprint.iacr.org/2017/1066.pdf"> "Bulletproofs: Short Proofs for Confidential Transactions and More" </a> by Benedikt Bunz, Jonathan Bootle, Dan Boneh et al: Original Bulletproofs paper. </li>
<li> <a href="https://rareskills.io/post/zk-multiplication"> "Zero Knowledge Multiplication" </a> by RareSkills: Explanation of Zero-Knowledge Multiplication protocol. </li>
</ul>
</div>
</div>
<!-- Lecture 13 -->
<div class="lecture-card">
<div class="lecture-card-head">
<div class="lecture-card-preview">
<img src="images/lecture-previews/lecture_13.png" class="preview">
</div>
<div class="lecture-card-content">
<div class="lecture-card-header">
<h2 class="lecture-card-title"> Lecture 13: Number Theoretic Transform (NTT/FFT) </h2>
<p class="lecture-card-speaker"> <strong>Speaker:</strong> Dmytro Zakharov </p>
</div>
<div class="lecture-card-content">
<p class="lecture-card-description">
<strong>Content.</strong> All previous lectures required (a) interpolation and (b) fast operations over polynomials. In this lecture, we consider the widely used technique to reduce the complexity of such tasks from quadratic to quasilinear. Namely, we consider:
<ul>
<li> Barycentric Interpolation. </li>
<li> Finite Field roots of unity. </li>
<li> What are NTT and Inverse NTT problems. </li>
<li> NTT algorithm derivation and implementation. </li>
</ul>
</p>
</div>
</div>
</div>
<div class="lecture-links-section">
<h1 class="lecture-links-header"> <strong> Lecture Material: </strong> </h1>
<div class="lecture-links-list">
<a href="files/notes/13-ntt.pdf" class="lecture-link">
<img src="images/icons/lecture_notes_icon.png">
<p class="link-name"> Lecture Notes </p>
</a>
<a href="files/slides/13-ntt.pdf" class="lecture-link">
<img src="images/icons/slides_icon.png">
<p class="link-name"> Slides </p>
</a>
<!-- <a href="https://github.com/ZKDL-Camp/circom-walkthrough" class="lecture-link">
<img src="images/icons/github.png">
<p class="link-name"> GitHub Repo </p>
</a> -->
</div>
<h1 class="lecture-links-header"> <strong> Other Helpful Resources: </strong> </h1>
<ul>
<li> <a href="https://youtu.be/h7apO7q16V0?si=h-CvEA6DREMsvxGC"> "The Fast Fourier Transform (FFT): Most Ingenious Algorithm Ever?" </a> by Reducible: Incredible Visual Explanation of the FFT algorithm (for real field case). </li>
<li> <a href="https://eprint.iacr.org/2024/585.pdf"> "A Complete Beginner Guide to the Number Theoretic Transform (NTT)" </a> by Ardianto Satriawan, Rella Mareta, and Hanho Lee: As the name suggests, a good tutorial on NTT with examples. </li>
<li> <a href="https://www.cs.rug.nl/~ando/pdfs/Ando_Emerencia_multiplying_huge_integers_using_fourier_transforms_paper.pdf"> "Multiplying Huge Integers using Fourier Transforms" </a> by Ando Emerencia: Well-explained tutorial on FFT based on large integer multiplication. </li>
<li> <a href="https://hackmd.io/@vbuterin/barycentric_evaluation"> "A Quick Barycentric Evaluation Tutorial" </a> by Vitalik Buterin: Explanation of Barycentric Interpolation derivation for both general case and roots of unity domain. </li>
</ul>
</div>
</div>
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<h2 class="lecture-card-title"> Lecture 12: PlonK Arithmetization </h2>
<p class="lecture-card-speaker"> <strong>Speaker:</strong> Nikita Masych </p>
</div>
<div class="lecture-card-content">
<p class="lecture-card-description">
<strong>Content.</strong> Groth16 is not the only zk-SNARK protocol used in practice! In this lecture, we will consider the PlonK protocol, which is frequently used in various zkEVM implementations. Here, we cover:
<ul>
<li> PlonK Arithmetization </li>
<li> How gates work in PlonK. Custom Gates. </li>
<li> Gate and Wiring Satisfiability Checks. </li>
<li> Permutation Check over Polynomials. </li>
</ul>
</p>
</div>
</div>
</div>
<div class="lecture-links-section">
<h1 class="lecture-links-header"> <strong> Lecture Material: </strong> </h1>
<div class="lecture-links-list">
<a href="files/notes/12-plonk-arithm.pdf" class="lecture-link">
<img src="images/icons/lecture_notes_icon.png">
<p class="link-name"> Lecture Notes </p>
</a>
<a href="files/slides/12-plonk-arithm.pdf" class="lecture-link">
<img src="images/icons/slides_icon.png">
<p class="link-name"> Slides </p>
</a>
<!-- <a href="https://github.com/ZKDL-Camp/circom-walkthrough" class="lecture-link">
<img src="images/icons/github.png">
<p class="link-name"> GitHub Repo </p>
</a> -->
</div>
<h1 class="lecture-links-header"> <strong> Other Helpful Resources: </strong> </h1>
<ul>
<li> <a href="https://eprint.iacr.org/2019/953.pdf"> "PlonK: Permutations over Lagrange-bases for Oecumenical Noninteractive arguments of Knowledge" </a> by Ariel Gabizon et. al.: original paper. Hard to read, but definitely worth it. </li>
<li> <a href="https://blog.lambdaclass.com/all-you-wanted-to-know-about-plonk/"> "All you wanted to know about Plonk" </a> by LambdaClass: Probably the most complete and well-explained article on PlonK anatomy. </li>
<li> <a href="https://vitalik.eth.limo/general/2019/09/22/plonk.html"> "Understanding PLONK" </a> by Vitalik Buterin: Another well-explained and illustrated guide on PlonK. </li>
</ul>
</div>
</div>
<!-- Lecture 11 -->
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<h2 class="lecture-card-title"> Lecture 11: Programming ZKPs in Circom </h2>
<p class="lecture-card-speaker"> <strong>Speaker:</strong> Kyrylo Riabov </p>
</div>
<div class="lecture-card-content">
<p class="lecture-card-description">
<strong>Content.</strong> After covering the Groth16 zk-SNARK theory, we will move to the practical part of the course. Here, we will learn how to use Circom to create circuits and generate proofs. Here, we cover:
<ul>
<li> Basic Circom syntax. </li>
<li> Creating circuits. </li>
<li> Generating proofs. </li>
<li> Interpreting all the generated data such as .r1cs, .sym, proof.json, and how they relate to the previously covered theory. </li>
</ul>
</p>
</div>
</div>
</div>
<div class="lecture-links-section">
<h1 class="lecture-links-header"> <strong> Lecture Material: </strong> </h1>
<div class="lecture-links-list">
<a href="files/notes/11-circom.pdf" class="lecture-link">
<img src="images/icons/lecture_notes_icon.png">
<p class="link-name"> Lecture Notes </p>
</a>
<a href="files/slides/11-circom.pdf" class="lecture-link">
<img src="images/icons/slides_icon.png">
<p class="link-name"> Slides </p>
</a>
<a href="https://github.com/ZKDL-Camp/circom-walkthrough" class="lecture-link">
<img src="images/icons/github.png">
<p class="link-name"> GitHub Repo </p>
</a>
</div>
<h1 class="lecture-links-header"> <strong> Other Helpful Resources: </strong> </h1>
<ul>
<li> <a href="https://ieeexplore.ieee.org/document/10002421"> "Circom: A Circuit Description Language for Building Zero-Knowledge Applications" </a> by Marta Belles-Munoz et al: Original Circom language paper. </li>
<li> <a href="https://zkintro.com/articles/programming-zkps-from-zero-to-hero"> "Programming ZKPs: From Zero to Hero" </a> by zkintro: Well-explained tutorial on working with Circom. </li>
<li> <a href="https://circom.erhant.me/index.html"> "Circom101" </a> by erhant: introduces many practical examples (arithmetic, comparators, hashing, merkle trees etc.) where Circom can be used. </li>
<li> <a href="https://www.rareskills.io/post/circom-tutorial"> "Circom language tutorial with circomlib walkthrough" </a> by Team RareSkills: a very detailed tutorial on how to use Circom with practical examples. </li>
</ul>
</div>
</div>
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<h2 class="lecture-card-title"> Lecture 10: Pairing-based zk-SNARKs </h2>
<p class="lecture-card-speaker"> <strong>Speakers:</strong> Anton Levochko and Dmytro Zakharov </p>
</div>
<div class="lecture-card-content">
<p class="lecture-card-description">
<strong>Content.</strong> Finally, we consider one of the most advanced zk-SNARKs: Pinocchio and Groth16. Here, we cover:
<ul>
<li> Turning QAP into succint verification over encrypted space. </li>
<li> Making SNARK sound. </li>
<li> Turning SNARK into zk-SNARK. </li>
<li> Pinocchio Protocol. </li>
<li> Groth16 Protocol. </li>
</ul>
</p>
</div>
</div>
</div>
<div class="lecture-links-section">
<h1 class="lecture-links-header"> <strong> Lecture Material: </strong> </h1>
<div class="lecture-links-list">
<a href="files/notes/10-groth16.pdf" class="lecture-link">
<img src="images/icons/lecture_notes_icon.png">
<p class="link-name"> Lecture Notes </p>
</a>
<a href="files/slides/10-groth16.pdf" class="lecture-link">
<img src="images/icons/slides_icon.png">
<p class="link-name"> Slides </p>
</a>
<!-- <a href="files/exercises/8-circuits.pdf" class="lecture-link">
<img src="images/icons/test.png">
<p class="link-name"> Task </p>
</a>
<a href="https://github.com/ZKDL-Camp/lecture-8-r1cs-qap" class="lecture-link">
<img src="images/icons/github.png">
<p class="link-name"> GitHub Repo </p>
</a> -->
</div>
<h1 class="lecture-links-header"> <strong> Other Helpful Resources: </strong> </h1>
<ul>
<li> <a href="https://arxiv.org/abs/1906.07221"> "Why and How zk-SNARK Works" </a> by Maksym Petkus: Very easy-to-follow tutorial on how zk-SNARKs work from scratch. Very friendly guide for beginners. </li>
<li> <a href="https://zk-learning.org/"> ZK MOOC, Spring 2023 </a>, "Linear PCP" lecture: amazing lecture on the idea of building Pinocchio and Groth16 protocols, expressed formally and mathematically. </li>
<li> <a href="https://eprint.iacr.org/2013/279.pdf"> "Pinocchio: Nearly Practical Verifiable Computation" </a> by Bryan Parno and Craig Gentry: original Pinocchio paper. Quite easy to understand if you are familiar with notation from lecture notes. </li>
<li> <a href="https://eprint.iacr.org/2016/260.pdf"> "On the Size of Pairing-based Non-interactive Arguments" </a> by Jens Groth: original Groth16 paper. A bit more formalistic view of the Groth16 protocol and what linear non-interactive proofs are used for. </li>
</ul>
</div>
</div>
<!-- Lecture 9 -->
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<h2 class="lecture-card-title"> Lecture 9: Quadratic Arithmetic Program. Probabilistically Checkable Proofs </h2>
<p class="lecture-card-speaker"> <strong>Speaker:</strong> Anton Levochko </p>
</div>
<div class="lecture-card-content">
<p class="lecture-card-description">
<strong>Content.</strong> With R1CS in our hands, we are now ready to succintly represent it as a Quadratic Arithmetic Program (QAP). Additionally, we consider other basic theory before specifying the Groth16 protocol. Here, we cover:
<ul>
<li> Turning R1CS into QAP. </li>
<li> What is PCP, IPCP, IOP. </li>
<li> Proof of Exponent. </li>
</ul>
</p>
</div>
</div>
</div>
<div class="lecture-links-section">
<h1 class="lecture-links-header"> <strong> Lecture Material: </strong> </h1>
<div class="lecture-links-list">
<a href="files/notes/9-qap-pcp.pdf" class="lecture-link">
<img src="images/icons/lecture_notes_icon.png">
<p class="link-name"> Lecture Notes </p>
</a>
<a href="files/slides/9-qap-pcp.pdf" class="lecture-link">
<img src="images/icons/slides_icon.png">
<p class="link-name"> Slides </p>
</a>
<!-- <a href="files/exercises/8-circuits.pdf" class="lecture-link">
<img src="images/icons/test.png">
<p class="link-name"> Task </p>
</a>
<a href="https://github.com/ZKDL-Camp/lecture-8-r1cs-qap" class="lecture-link">
<img src="images/icons/github.png">
<p class="link-name"> GitHub Repo </p>
</a> -->
</div>
<h1 class="lecture-links-header"> <strong> Other Helpful Resources: </strong> </h1>
<ul>
<li> <a href="https://arxiv.org/abs/1906.07221"> "Why and How zk-SNARK Works" </a> by Maksym Petkus: Very easy-to-follow tutorial on how zk-SNARKs work from scratch. Very friendly guide for beginners. </li>
<li> <a href="https://medium.com/@VitalikButerin/quadratic-arithmetic-programs-from-zero-to-hero-f6d558cea649"> "Quadratic Arithmetic Programs: From Zero to Hero" </a> by Vitalik Buterin: R1CS and QAP demonstrated over real numbers. </li>
</ul>
</div>
</div>
<!-- Lecture 8 -->
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<div class="lecture-card-header">
<h2 class="lecture-card-title"> Lecture 8: SNARKs. Arithmetical Circuits </h2>
<p class="lecture-card-speaker"> <strong>Speaker:</strong> Anton Levochko </p>
</div>
<div class="lecture-card-content">
<p class="lecture-card-description">
<strong>Content.</strong> This is an opening lecture on SNARKs: technology we are using daily in our projects. Here, we cover:
<ul>
<li> The definition of zk-SNARK. </li>
<li> Arithmetic Circuits. Circuit Satisfability Problem. </li>
<li> Rank-1 Constraint System (R1CS) in vector and matrix forms. </li>
</ul>
</p>
</div>
</div>
</div>
<div class="lecture-links-section">
<h1 class="lecture-links-header"> <strong> Lecture Material: </strong> </h1>
<div class="lecture-links-list">
<a href="files/notes/8-circuits.pdf" class="lecture-link">
<img src="images/icons/lecture_notes_icon.png">
<p class="link-name"> Lecture Notes </p>
</a>
<a href="files/slides/8-circuits.pdf" class="lecture-link">
<img src="images/icons/slides_icon.png">
<p class="link-name"> Slides </p>
</a>
<a href="files/exercises/8-circuits.pdf" class="lecture-link">
<img src="images/icons/test.png">
<p class="link-name"> Task </p>
</a>
<a href="https://github.com/ZKDL-Camp/lecture-8-r1cs-qap" class="lecture-link">
<img src="images/icons/github.png">
<p class="link-name"> GitHub Repo </p>
</a>
</div>
<h1 class="lecture-links-header"> <strong> Other Helpful Resources: </strong> </h1>
<ul>
<li> <a href="https://arxiv.org/abs/1906.07221"> "Why and How zk-SNARK Works" </a> by Maksym Petkus: Very easy-to-follow tutorial on how zk-SNARKs work from scratch. Very friendly guide for beginners. </li>
<li> <a href="https://www.rareskills.io/post/rank-1-constraint-system"> "Converting Algebraic Circuits to R1CS" </a> by RareSkills: Also well-explained how R1CS works with many practical examples. </li>
</ul>
</div>
</div>
<!-- Lecture 7 -->
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<h2 class="lecture-card-title"> Lecture 7: Sigma Protocols </h2>
<p class="lecture-card-speaker"> <strong>Speaker:</strong> Dmytro Zakharov </p>
</div>
<div class="lecture-card-content">
<p class="lecture-card-description">
<strong>Content.</strong> Here, we will consider the most basic form of interactive proofs - Sigma Protocols. We will understand how to turn them into non-interactive proofs and how to use them in practice. Here, we cover:
<ul>
<li> Schnorr Signature Scheme. </li>
<li> Sigma Protocols. </li>
<li> Examples of Sigma Protocols: Okamoto's and Chaum-Pedersen Protocols </li>
<li> Combining Sigma Protocols. </li>
</ul>
</p>
</div>
</div>
</div>
<div class="lecture-links-section">
<h1 class="lecture-links-header"> <strong> Lecture Material: </strong> </h1>
<div class="lecture-links-list">
<a href="files/notes/7-sigma.pdf" class="lecture-link">
<img src="images/icons/lecture_notes_icon.png">
<p class="link-name"> Lecture Notes </p>
</a>
<a href="files/slides/7-sigma.pdf" class="lecture-link">
<img src="images/icons/slides_icon.png">
<p class="link-name"> Slides </p>
</a>
<a href="files/exercises/7-sigma.pdf" class="lecture-link">
<img src="images/icons/test.png">
<p class="link-name"> Exercises </p>
</a>
<a href="https://forms.gle/9vW5viQSunms62bD8" class="lecture-link">
<img src="images/icons/google_form.png">
<p class="link-name"> Test Form </p>
</a>
<a href="https://github.com/ZKDL-Camp/lecture-7-sigma" class="lecture-link">
<img src="images/icons/github.png">
<p class="link-name"> GitHub Repo </p>
</a>
</div>
<h1 class="lecture-links-header"> <strong> Other Helpful Resources: </strong> </h1>
<ul>
<li> <a href="https://toc.cryptobook.us/"> "A Graduate Course in Applied Cryptography" </a> by Dan Boneh and Victor Shoup, <strong>Section 19</strong>: Very formalistic and rigorous introduction to Sigma Protocols and turning interactiveness to non-interactiveness in the subsequent <strong>Section 20</strong>. </li>
<li> <a href="https://people.cs.georgetown.edu/jthaler/ProofsArgsAndZK.pdf"> "Proofs, Arguments, and Zero-Knowledge" </a> by Justin Thaler, <strong> Chapter 12 </strong>: quite rigorous description of Discrete Logarithm-based zero-knowledge protocols. </li>
</ul>
</div>
</div>
<!-- Lecture 6 -->
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<div class="lecture-card-header">
<h2 class="lecture-card-title"> Lecture 6: Introduction to Zero-Knowledge Proofs </h2>
<p class="lecture-card-speaker"> <strong>Speaker:</strong> Dmytro Zakharov </p>
</div>
<div class="lecture-card-content">
<p class="lecture-card-description">
<strong>Content.</strong> This lecture finally introduces the concept of Zero-Knowledge Proofs and their applications. Here, we cover:
<ul>
<li> What is a cryptographic proof exactly? </li>
<li> Interactive Proofs. </li>
<li> Soundness and Zero-Knowledge definitions. </li>
<li> Proof vs Proof of Knowledge. </li>
<li> Fiat-Shamir Heuristic. </li>
</ul>
</p>
</div>
</div>
</div>
<div class="lecture-links-section">
<h1 class="lecture-links-header"> <strong> Lecture Material: </strong> </h1>
<div class="lecture-links-list">
<a href="files/notes/6-intro-zk.pdf" class="lecture-link">
<img src="images/icons/lecture_notes_icon.png">
<p class="link-name"> Lecture Notes </p>
</a>
<a href="files/slides/6-intro-zk.pdf" class="lecture-link">
<img src="images/icons/slides_icon.png">
<p class="link-name"> Slides </p>
</a>
<a href="files/exercises/6-intro-zk.pdf" class="lecture-link">
<img src="images/icons/test.png">
<p class="link-name"> Exercises </p>
</a>
<a href="https://forms.gle/h3hrk4ug6B5LAgLs6" class="lecture-link">
<img src="images/icons/google_form.png">
<p class="link-name"> Test Form </p>
</a>
</div>
<h1 class="lecture-links-header"> <strong> Other Helpful Resources: </strong> </h1>
<ul>
<li> <a href="https://zk-learning.org/"> "Zero Knowledge Proofs MOOC, Spring 2023" </a> Amazing series of lectures on zero-knowledge proofs. This lecture was primarily based on the first lecture by Shafi Goldwasser. </li>
<li> <a href="https://www.cs.princeton.edu/courses/archive/fall07/cos433/lec15.pdf"> "Zero Knowledge Proofs Lecture" </a> by Boaz Barak: very well-explained fundamentals of zero-knowledge proof with a detailed analysis of a quadratic residue interactive protocol. </li>
</ul>
</div>
</div>
<!-- Lecture 5 -->
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<h2 class="lecture-card-title"> Lecture 5: Cryptographic commitment schemes </h2>
<p class="lecture-card-speaker"> <strong>Speaker:</strong> Denis Riabtsev </p>
</div>
<div class="lecture-card-content">
<p class="lecture-card-description">
<strong>Content.</strong> In this lecture we will dive into the design and application of various cryptographic commit schemes that are often used in zero knowledge proof systems. All in all, here we cover:
<ul>
<li> Hash-based commitments. </li>
<li> Vector commitments. </li>
<li> Polynomial commitments. </li>
</ul>
</p>
</div>
</div>
</div>
<div class="lecture-links-section">
<h1 class="lecture-links-header"> <strong> Lecture Material: </strong> </h1>
<div class="lecture-links-list">
<a href="files/notes/5-commitments.pdf" class="lecture-link">
<img src="images/icons/lecture_notes_icon.png">
<p class="link-name"> Lecture Notes </p>
</a>
<a href="files/slides/5-commitments.pdf" class="lecture-link">
<img src="images/icons/slides_icon.png">
<p class="link-name"> Slides </p>
</a>
<a href="files/exercises/5-commitments.pdf" class="lecture-link">
<img src="images/icons/test.png">
<p class="link-name"> Exercises </p>
</a>
<a href="https://docs.google.com/forms/d/e/1FAIpQLSfBYGvraga2D2jGWy5vuJ46Uy6a4sYb_A33FSmFhnq2E_Cr5g/viewform" class="lecture-link">
<img src="images/icons/google_form.png">
<p class="link-name"> Test Form </p>
</a>
</div>
<h1 class="lecture-links-header"> <strong> Other Helpful Resources: </strong> </h1>
<ul>
<li> <a href="https://docs.scroll.io/en/learn/zero-knowledge/kzg-commitment-scheme/"> KZG Commitment Scheme </a> by <i>Scroll</i>, explains the math behind KZG commitment schemes. </li>
<li> <a href="https://assets.super.so/9c1ce0ba-bad4-4680-8c65-3a46532bf44a/files/61fb28e6-f2dc-420f-89e1-cc8000233a4f.pdf"> Modern Zero Knowledge Cryptography. Lecture 5 </a> by Ying Tong with formally explained vector and polynomial commitments. </li>
<li> <a href="https://www.rareskills.io/post/pedersen-commitment"> "What are Pedersen Commitments and How They Work" </a> by <i>RareSkills</i>: an article that explains Pedersen commitments in simple words with code examples in Python. </li>
</ul>
</div>
</div>
<!-- Lecture 4 -->
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<h2 class="lecture-card-title"> Lecture 4: Projective Coordinates and Pairing </h2>
<p class="lecture-card-speaker"> <strong>Speaker:</strong> Dmytro Zakharov </p>
</div>
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<strong>Content.</strong> 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:
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<li> Relations and equivalence classes. </li>
<li> Projective Coordinates. </li>
<li> Adding points in projective coordinates. Scalar multiplication using double-and-add algorithm. </li>
<li> Elliptic Curve Pairing. </li>
<li> Pairing applications. </li>
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<h1 class="lecture-links-header"> <strong> Lecture Material: </strong> </h1>
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<a href="files/notes/4-pairing.pdf" class="lecture-link">
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<p class="link-name"> Lecture Notes </p>
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<a href="files/slides/4-pairing.pdf" class="lecture-link">
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<p class="link-name"> Slides </p>
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<a href="files/exercises/4-pairing.pdf" class="lecture-link">
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<p class="link-name"> Exercises </p>
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<a href="https://forms.gle/3Bq4DTWkp53dtdyA9" class="lecture-link">
<img src="images/icons/google_form.png">
<p class="link-name"> Test Form </p>
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<a href="https://github.com/ZKDL-Camp/lecture-4-pairings" class="lecture-link">
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<p class="link-name"> GitHub Repo </p>
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<h1 class="lecture-links-header"> <strong> Other Helpful Resources: </strong> </h1>
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<li> <a href="https://static1.squarespace.com/static/5fdbb09f31d71c1227082339/t/5ff394720493bd28278889c6/1609798774687/PairingsForBeginners.pdf"> "Pairing for Beginners" </a> by Craig Costello: Not only this already mentioned book contains pairing inner working, but also a profound introduction into projective form of elliptic curves. </li>
<li> <a href="https://toc.cryptobook.us/"> "A Graduate Course in Applied Cryptography" </a> by Dan Boneh and Victor Shoup, <strong>Section 15</strong>. Recommended to read this chapter up to section 15.5 including. </li>
<li> <a href="https://hackmd.io/@jpw/bn254"> "BN254 For The Rest of Us" </a> by Jonathan Wang: if you want specifics of pairing-friendly curve parameters, you can take a look here. This curve is currently widely used. </li>
<li> <a href="https://eprint.iacr.org/2010/354.pdf"> High-Speed Software Implementation of the Optimal Ate Pairing over Barreto-Naehrig Curves </a> by Jean-Luc Beuchat et al.: if you <strong>really</strong> want to get into the details of pairing implementation, this paper is the best. It contains all the algorithms to implement pairing from scratch. </li>
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<h2 class="lecture-card-title"> Lecture 3: Finite Field Extensions and Elliptic Curves </h2>
<p class="lecture-card-speaker"> <strong>Speaker:</strong> Dmytro Zakharov </p>
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<strong>Content.</strong> Our primary focus will be on Finite Field Extensions, while also covering basics of Elliptic Curves. Here, we cover:
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<li> Finite Field Extensions </li>
<li> Algebraic Closure </li>
<li> Elliptic Curve Definition </li>
<li> Discere Logarithm on Elliptic Curves </li>
</ul>
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<h1 class="lecture-links-header"> <strong> Lecture Material: </strong> </h1>
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<a href="files/notes/3-fields-and-ec.pdf" class="lecture-link">
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<p class="link-name"> Lecture Notes </p>
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<a href="files/slides/3-ec.pdf" class="lecture-link">
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<p class="link-name"> Slides </p>
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<a href="files/exercises/3-ec.pdf" class="lecture-link">
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<p class="link-name"> Exercises </p>
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<a href="https://forms.gle/LiXaeXkzzHFyLvMX9" class="lecture-link">
<img src="images/icons/google_form.png">
<p class="link-name"> Test Form </p>
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<h1 class="lecture-links-header"> <strong> Other Helpful Resources: </strong> </h1>
<ul>
<li> <a href="https://www.math.uchicago.edu/~may/VIGRE/VIGRE2009/REUPapers/Moy.pdf"> An Introduction to the Theory of Field Extensions </a> 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. </li>
<li> <a href="http://tomlr.free.fr/Math%E9matiques/Math%20Complete/Cryptography/Guide%20to%20Elliptic%20Curve%20Cryptography%20-%20D.%20Hankerson,%20A.%20Menezes,%20S.%20Vanstone.pdf"> "Guide to Elliptic Curve Cryptography" </a> by Alfred Menezes et al.: Amazing book on Elliptic Curves optimizations. It also contains a great introduction into finite fields. </li>
<li> <a href="https://static1.squarespace.com/static/5fdbb09f31d71c1227082339/t/5ff394720493bd28278889c6/1609798774687/PairingsForBeginners.pdf"> "Pairing for Beginners" </a> by Craig Costello: Quick dive into inner workings of elliptic curve pairing. </li>
<li> <a href="https://toc.cryptobook.us/"> "A Graduate Course in Applied Cryptography" </a> by Dan Boneh and Victor Shoup, <strong>Section 15</strong> (Elliptic curve cryptography and pairings): Elliptic Curves are described very well together with the basics of pairing (without going into the details of construction). </li>
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