Elementary Bitcoin

Preface

This is a mathematics textbook about Bitcoin, written in the classical idiom of numbered definitions, theorems, proofs, and exercises: across forty chapters and four appendices, 355 definitions, nearly one hundred theorems and propositions, and 171 exercises. Every result is proved before it is used, from the group axioms through secp256k1, ECDSA, and Schnorr, to SPV, compact filters, Lightning, Taproot, covenants, and the consensus cleanup.

We assume no prior knowledge of cryptography or of Bitcoin. We do assume mathematical maturity: comfort reading a proof, checking a computation, and holding a definition to its exact wording. A patient reader with school algebra can build that comfort here; Volume I was written for exactly that purpose. But the proofs are the book, not an ornament to it.

This is not an engineering manual, and it contains no code: for building software, Antonopoulos's Mastering Bitcoin and Song's Programming Bitcoin are the standard references and natural complements. Nor is it a beginner's explainer. Its nearest relative is Narayanan et al.'s Bitcoin and Cryptocurrency Technologies (Princeton, 2016), from which it differs in being current, Bitcoin-only, and written theorem-first. Each concept builds on those before it, forming a complete chain of understanding: the reader who works through these volumes can verify, from first principles, every cryptographic claim made about Bitcoin.

A Note on Epistemic Status

Not everything in this book is true in the same way, and the reader deserves to know which kind of truth each part offers. Volume I is mathematics: its theorems are proved and will be true in a century. Volumes II and III are verifiable protocol fact—every claim can be checked against running code and the cited specifications, and changes only when the protocol does. Volume IV surveys the contested present: fork histories whose narratives are still argued, proposals whose statuses drift, governance questions that are contested by construction. There the book reports positions, attributes them to their holders, and dates every perishable claim. Volume V is disciplined speculation—economic models with stated assumptions and scenarios labeled as scenarios, kept in remarks and never dressed as theorems. (In those later volumes, definition boxes fix terminology rather than ground proofs.)

The gradient is the design. The first three volumes give you the tools; the last two show you the open questions those tools illuminate—and by the time you reach them, you will not need this book to tell you what to think.

How to Read This Book

The chapters are ordered so that everything is proved before it is used: that is what "elementary" means here—self-contained, not simple. But the book does not have to be read front to back, and most of the later volumes stand on a small set of prerequisites. Four paths:

The full course. Chapters 1–40 in order. Volume I is the investment; everything after it is payoff, with no claim left unproved.

Bitcoin first. Start at Chapter 9 (keys and addresses) and read through Chapter 16, taking the cryptography of Volume I on faith—the forward references will tell you exactly which proofs you deferred. Return to Chapters 1–8 when you want to own them.

Verification and scaling. Chapters 12–13 (Merkle trees and blocks), then Volume III (Chapters 17–25). This is the path for evaluating SPV, light clients, and layer-2 claims—and the debates of Chapter 25 are its destination.

The debates and the future. Chapters 25–27 (myths and fork history), 33 (governance), and 38–40 (security budget, quantum, monetary future) are largely self-contained prose and can be read first by anyone who wants to know what the arguments are about before studying the machinery beneath them.

Wherever you start, Appendix A (notation), Appendix C (subject index), and Appendix D (the rule catalog) are designed for random access.

About This Project

Elementary Bitcoin is written by Melvin Carvalho and maintained as a community project: the source is public, the text is licensed CC BY-SA 4.0, and corrections and contributions flow through GitHub. This edition is dated June 2026; Volume IV dates every perishable claim in place. The numerical examples have been verified computationally (curve points recomputed, signatures checked, probability tables re-derived) and all of its figures rendered and inspected.

Some subjects are deliberately out of scope: custody operations, exchange engineering, price models, and implementation internals. Others are planned, and contributions toward them are especially welcome: mempool and fee policy, wallet mathematics (BIP-32, descriptors, miniscript), mining pool economics, on-chain privacy, and the peer-to-peer network layer.

The claims of this book are checkable at every level: proofs can be re-derived, protocol facts checked against running code and the cited BIPs, attributed positions verified against their sources. An error found at any level is a contribution. So is disagreement with the analysis.