The Shared Math Under Every Blockchain

Bitcoin, Ethereum, Zcash and Monero look like rival species, but underneath they share one skeleton of mathematics. Almost every cryptocurrency ever shipped is assembled from the same four mathematical ideas: modular arithmetic over finite fields, groups and the discrete-logarithm problem, one-way hash functions, and digital signatures built from the first three. Learn this shared toolkit once and every chain becomes a different arrangement of the same parts. This course teaches that toolkit, then shows exactly where Monero arranges it differently to buy privacy.

Finite Fields: Arithmetic That Wraps Around

Cryptography never works with ordinary unbounded integers — it works modulo a fixed number, so values wrap around like a clock. A finite field 𝔽_p is the set {0, 1, …, p−1} with addition and multiplication done mod a prime p. Because p is prime, every non-zero element has a multiplicative inverse, so you can also "divide". This is the sandbox all the heavy machinery plays in. Bitcoin's curve lives over a field of size p = 2²⁵⁶ − 2³² − 977; Monero's lives over q = 2²⁵⁵ − 19. Different primes, same idea: a closed universe of numbers where arithmetic never overflows to infinity.

Groups and the Discrete-Logarithm Problem

Stack a field into a group — a set with one operation you can repeat — and you get the engine of public-key cryptography. Pick a generator G and a secret integer x; computing x·G (the operation applied x times) is fast. Going backwards — recovering x from G and x·G — is the discrete-logarithm problem (DLP), and for the right group it is believed computationally infeasible. That one-way asymmetry is the whole trick:

• Your private key is the secret scalar x.
• Your public key is the point x·G — safe to publish, because nobody can invert it.

Every elliptic-curve coin — Bitcoin, Ethereum, Monero — is staking its entire security on the ECDLP being hard. They differ only in which group they use, not in the principle.

One-Way Hash Functions

A cryptographic hash H(x) shreds any input into a fixed-length fingerprint with three properties: it is deterministic, preimage-resistant (you can't run it backwards), and collision-resistant (you can't find two inputs with the same output). Hashes are the connective tissue of every blockchain — they chain blocks together, compress data into Merkle roots, derive keys, and, crucially for Monero, let you commit to a value without revealing it. Bitcoin leans on SHA-256; Ethereum and Monero use Keccak. We compare them next, in Hash Functions: The Common Workhorse.

Digital Signatures: The Three Combined

Put the group and the hash together and you can sign: prove you know the private scalar behind a public point without revealing the scalar, binding that proof to a specific message. A valid signature says "the holder of the key authorized exactly this transaction." Bitcoin and Ethereum use ECDSA; Monero uses an EdDSA-style construction generalized into a ring signature. Same primitive, profoundly different privacy — the subject of Digital Signatures.

Why This Framing Matters for Monero

Here is the thesis of the whole course: Monero invents almost no new mathematics. It uses the same finite fields, the same discrete-log hardness, the same hash-based commitments that secure Bitcoin. What it changes is what it hides. A transparent chain publishes the public keys and amounts directly on the ledger; Monero takes the identical building blocks and composes them so the sender, the receiver, and the amount are all provably correct yet never revealed. To see how, you first have to be fluent in the shared parts — so next we look closely at the first one, the humble hash function.