Finite Fields: The Hidden Math Powering Blockchains

#blockchain #cryptography #web3

Week 1: Finite Field Elements: the math quietly powering blockchains

A few weeks ago, I promised a public build challenge: we’d explore blockchain from the ground up in Go, week by week, sharing insights, code, and the intuition behind it all.

Here’s the first installment.

Before elliptic curves, digital signatures, or zero-knowledge proofs, blockchains rely on something much more fundamental: finite fields.

If this concept isn’t clear, cryptography feels like magic.
If it is clear, everything else starts to click.

What is a Finite Field?

Think of a clock.

On a 12 hour clock:

The numbers only go from 0 to 11
9 + 5 doesn’t give 14, it wraps around and gives 2

This is an example of modular arithmetic, a simple way to understand finite fields.

A finite field works similarly, but with some important differences:

The “clock size” (the number of elements) is usually a prime number or a power of a prime
Every non-zero number has a multiplicative inverse
This makes division always possible within the field

That last property is why primes are critical in cryptography: without inverses, many algorithms break.

Why Blockchains Care

Finite fields provide:

Deterministic arithmetic: every operation is predictable
Exact computation: no floating-point errors or rounding ambiguity
Guaranteed inverses: critical for cryptographic signatures and key operations

Elliptic curves, ECDSA, Schnorr, all of it is built on top of this.
If arithmetic isn’t perfectly predictable, cryptography collapses.

What’s a Field Element?

A field element is simply:

a number that lives inside a finite field

Its value is always interpreted modulo the field size.
For example, in GF(19), 7 and 26 are actually the same field element because 26 ≡ 7 mod 19.

Field elements are the building blocks of cryptography: they’re the “numbers” that elliptic curves, signatures, and blockchains operate on.

That’s exactly what my FieldElement type models: a value plus the field it belongs to.

Values always stay within the field
Operations only happen between elements of the same field

This design ensures invalid math is impossible to ignore. Errors show up early, instead of silently breaking cryptographic logic later. That’s intentional.