Introduction

This book is an ongoing effort to gather some notes about cryptography with a focus on schemes which are relevant to the decentralized web such as multiparty signatures, zero-knowledge proofs, etc.

For now it consists of the following parts:

• Mathematics:
• Cryptography:
• Proof Systems:
• ZK Hack Puzzles Walk-through:

We assume that the reader has some basic knowledge of arithmetic and algebra and of common concepts from cryptography (hash functions, signatures, ...).

Here are a number of freely available textbooks to learn more (we will point to specific sections of them when needed):

• A Computational Introduction to Number Theory and Algebra by Victor Shoup
• the Boneh-Shoup cryptography textbook
• lecture notes for an advanced graduate cryptography course by Jonathan Katz
• The Joy of Cryptography by Mike Rosulek
• Mathematics of Public Key Cryptography by Steven Galbraith
• Least Authority's MoonMath manual
• Proofs, Arguments, and Zero-Knowledge by Justin Thaler

Mathematical Notation

• Given a set $S,$ we let $S^n$ denote the set of strings of length $n$ over $S,$ and $S^{\ast }$ the set of all strings, i.e., $S^{\ast }={\cup }_{n\ge 0}{S}^n,$ where $S^0$ denotes the singleton consisting of the empty string; the length of a string $x$ is denoted $|x|.$
• Given a non-empty finite set $S,$ the sampling of a variable $s$ according to the uniform distribution is denoted $s{\leftarrow }_{\$}S.$
• Unless specified otherwise, groups are denoted additively.
• Main algebraic structures:

Note that all proofs throughout the book are collapsible:

Proof

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Acknowledgments

This book is built with mdBook using the following preprocessors:

• mdBook-KaTeX
• mdbook-toc
• mdbook-footnote
• mdbook-mermaid
• mdbook-admonish
• mdbook-numthm
• mdbook-numeq
• mdbook-mathpunc

If you spot anything off, I'd be happy to get your feedback and acknowledge it here.