BA-02 · Intermediate

Mathematics and cryptography

Understand the primitives without treating cryptography as magic.

Follow these 9 lessons in order, or choose the topic you need. Each lesson combines an explanation, a worked example and practice inside the Guild.

  1. Modular arithmetic

    Modular arithmetic wraps numbers around a fixed boundary, like hours on a clock. Bitcoin's signature mathematics uses this kind of arithmetic.

    35 minutes · Open lesson

  2. Hash functions and SHA-256

    A hash turns any message into a fixed-size fingerprint. Bitcoin commonly uses the Secure Hash Algorithm 256, called SHA-256, to detect changes and make commitments.

    45 minutes · Open lesson

  3. Hash160 and Merkle trees

    Bitcoin combines two hash functions for some identifiers and arranges transaction hashes in a tree so one item can be proven part of a larger set.

    55 minutes · Open lesson

  4. Elliptic-curve keys

    A private key is a secret number. Bitcoin multiplies it by a fixed curve point to create a public key, a one-way operation used to authorize spending.

    65 minutes · Open lesson

  5. ECDSA signatures

    The Elliptic Curve Digital Signature Algorithm, called ECDSA, lets a private key authorize a message without revealing the key. Safe signing also depends on a unique secret nonce.

    35 minutes · Open lesson

  6. Schnorr and BIP340

    Schnorr signatures use simpler algebra than Bitcoin's earlier signature system. Bitcoin Improvement Proposal 340 defines the exact Schnorr format used by Taproot.

    45 minutes · Open lesson

  7. Tagged hashes

    A tagged hash adds a purpose label before hashing. This stops the same bytes from being mistaken for a message used by a different protocol rule.

    55 minutes · Open lesson

  8. Threshold cryptography

    Threshold signing divides authority among several participants. A required number can create one valid group signature without combining their secret shares.

    65 minutes · Open lesson

  9. The Bitcoin cryptography map

    Connect SHA-256, HASH256, HASH160, HMAC-SHA512, PBKDF2, secp256k1, ECDSA, Schnorr, tagged hashes, and Merkle commitments to the exact job each performs.

    35 minutes · Open lesson