Bitcoin Dojo — Cryptography Fundamentals Track

A hands-on implementation of elliptic curve cryptography from first principles, written in Rust.

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Overview

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What I Built

This project walks through the full cryptographic stack that powers Bitcoin — built from scratch, layer by layer.

Module	Topic	What It Does
1.1	Hashing & Randomness	SHA-256 hashing and cryptographically secure random number generation
1.2	Field Element	Arithmetic over a finite field — numbers mod a prime p
1.3	Scalar	Arithmetic mod the curve order n — the space where private keys live
1.4	Curve Point	Elliptic curve point addition and the double-and-add scalar multiplication algorithm
1.5	Secp256k1	The specific curve parameters used by Bitcoin (p, n, G)
1.6	Keys	Private key and public key generation (Q = d * G)
1.7	ECDSA	Elliptic Curve Digital Signature Algorithm — sign and verify messages

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Key Concepts Covered

Modular Arithmetic

Finite Fields

Fermat's Little Theorem

Mathematical Groups

Elliptic Curves

Point Addition

Double-and-Add Algorithm

Public Key Cryptography

ECDSA

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Project Structure

bitcoin_dojo/
├── src/
│   ├── lib.rs
│   └── ecc/
│       ├── constants.rs    # secp256k1 curve parameters (p, n, G)
│       ├── field.rs        # FieldElement — arithmetic mod p
│       ├── scalar.rs       # Scalar — arithmetic mod n
│       ├── curve.rs        # Point — elliptic curve point operations
│       ├── keys.rs         # PrivateKey / PublicKey
│       ├── ecdsa.rs        # sign() / verify() with RFC 6979
│       └── util.rs         # SHA-256, secure random
└── tests/
    └── ecc/
        ├── field_tests.rs
        ├── scalar_tests.rs
        ├── curve_tests.rs
        ├── keys_tests.rs
        ├── ecdsa_tests.rs
        └── util_tests.rs

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Running the Tests

# Run all tests
cargo test

# Run tests for a specific module
cargo test field
cargo test scalar
cargo test curve
cargo test ecdsa

# Run with output visible
cargo test -- --nocapture

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The Cryptographic Stack

┌─────────────────────────────────────────┐
│  1.7  ECDSA — sign() / verify()         │
├─────────────────────────────────────────┤
│  1.6  Keys — PrivateKey / PublicKey     │
├─────────────────────────────────────────┤
│  1.5  Secp256k1 — p, n, G constants     │
├─────────────────────────────────────────┤
│  1.4  Curve Point — add, double-and-add │
├─────────────────────────────────────────┤
│  1.3  Scalar — arithmetic mod n         │
├─────────────────────────────────────────┤
│  1.2  Field Element — arithmetic mod p  │
├─────────────────────────────────────────┤
│  1.1  Hashing & Randomness              │
└─────────────────────────────────────────┘

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Dependencies

[dependencies]
num-bigint = "0.4"
lazy_static = "1.4"
sha2 = "0.10"
hmac = "0.12"
rand = "0.9"
hex = "0.4"

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Key Takeaways

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Real-World Applications

• Bitcoin / Ethereum transaction signing
• TLS/HTTPS certificate authentication
• SSH key-based authentication
• JWT token signing (ES256)
• Code signing and software verification
• Hardware wallet key management
• Zero-knowledge proof systems

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References

• Programming Bitcoin — Jimmy Song
• Bitcoin Whitepaper — Satoshi Nakamoto
• SEC 2: Recommended Elliptic Curve Domain Parameters
• RFC 6979 — Deterministic ECDSA

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Author

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License