On Entropy — Probability, Randomness, and the Safety of a Bitcoin Key

Front Matter

PPrologue: The Weakest Link

A 256-bit key is unguessable — if it is actually random. The method and the road.

Part I — Chance

1Counting

Outcomes, arrangements, and the multiplication principle — including the twelve-word question.

2Probability

From counting to chances: independence, complements, and reading “1 in 2ⁿ” honestly.

3Expectation and the Long Run

Expected value and the law of large numbers — why a small chance per guess demands many guesses.

Part II — Entropy

4Information as Surprise

Bits as answers to yes-or-no questions; Shannon entropy built from counting.

5The Attacker's Metric

Min-entropy and guessing: the measure that matters when one guess steals everything.

6Where Randomness Comes From

Physical noise, OS entropy pools, PRNG vs CSPRNG, and dice.

Part III — The Key

7A 256-Bit Number

What a Bitcoin private key is, and the size of the space it hides in.

8Guessing Games at Scale

Brute-force physics and keyspace arithmetic — what an attack would actually cost.

Part IV — When Randomness Fails

9Low-Entropy Keys in the Wild

Brainwallet drains, KeyBleed, and mempool fee races over weak keys.

10The Nonce Catastrophe

One leaked nonce loses the key — and the deterministic fix.

Part V — Protecting the Key

11Seed Phrases

BIP 39: entropy to checksum to words — and why you never rearrange them.

12The Operational Checklist

Generate offline, hardware wallets, passphrases, backups: the practice.

Appendices & Reference

ANotation, Brick by Brick

Every symbol used in this book, built before it is used.

BTables: Powers of Two and Probability

Reference tables for counting, chance, and keyspace sizes.

CSpecimens and References

The real-world failures and sources behind the chapters.

Interactive Demos — Linkable & Shareable

⚡The Counting Playground

Choices multiply, live: stages, the doubling ladder, arrangements and choices in exact arithmetic. (Ch. 1)

⚡Two Measures of Surprise

Drag the chances and watch Shannon's average and the attacker's best guess move — the gap is the thief's discount. (Ch. 4 & 5)

⚡Roll Your Own Key

Coins, dice, and the 2048-word list: count the bits as you roll, and assemble a real draw in your browser. (Ch. 6)

⚡The Invoice

Price a brute-force search: rate, space, targets — and read the answer in universe-lifetimes. (Ch. 8)

⚡The Nonce Catastrophe, Step by Step

Two signatures, one reused nonce: walk the attacker's algebra and recover the key yourself. (Ch. 10)