This course condenses the essential mathematics behind Bitcoin’s cryptographic systems into a highly practical workflow. Concepts are explained, demonstrated with examples, and then implemented in Jupyter Notebook. The guiding idea is simple: you only truly understand a cryptographic primitive once you code it. Across the two-day structure, students generate testnet addresses, build and broadcast transactions, and eventually interact with the network without block explorers. All of this requires a solid foundation in finite fields and elliptic curves.

A finite field F(p) is an arithmetic system defined by a prime number p, containing elements 0 through p–1. Prime fields ensure every non-zero element has an inverse and all operations remain within the field. Arithmetic wraps around using modulo p for addition, subtraction, and multiplication. Python’s pow(base, exp, mod) enables efficient modular exponentiation, crucial for large exponents used in real cryptographic operations.

Multiplying any non-zero element k by all elements of a prime field produces a permutation of the field. This property guarantees uniformity and prevents structural weaknesses, making prime fields ideal for secure key generation and digital signatures.

Division is implemented via multiplicative inverses. Fermat’s Little Theorem states that
n^(p–1) ≡ 1 (mod p), so the inverse is n^(p–2). Python supports this directly with pow(n, -1, p). These operations appear constantly in ECDSA and Bitcoin’s underlying cryptographic routines.

Elliptic curves follow the equation y² = x³ + ax + b. Bitcoin uses secp256k1, defined as y² = x³ + 7 over a finite field. Points on an elliptic curve form a mathematical group under point addition. A line drawn through two points P and Q intersects the curve at a third point R; reflecting R across the x-axis yields P + Q. This system is associative and includes an identity element: the point at infinity.

Doubling a point uses a tangent line instead of a secant line, with slope derived from the curve’s derivative. Although these formulas involve calculus over real numbers, they become fully discrete and exact when the curve is defined over a finite field—the context used in Bitcoin.

Finite fields provide deterministic, invertible arithmetic; elliptic curves provide a nonlinear structure where computing k·P is easy but reversing it is computationally infeasible. Combining both yields the foundation for Bitcoin’s public/private keys, ECDSA signatures, and transaction security. Understanding these fundamentals prepares students to implement keys, transactions, and signatures directly—without abstractions or external tools.