# What is a finite field?

A finite field is a set with a fixed number of elements, where addition, subtraction, multiplication and division by a nonzero element always stay inside the set. STARKs do their arithmetic in one.

Also called Galois field.

## What a field is

A field is a set where addition, subtraction, multiplication and division by a nonzero element all work, and the usual rules hold, such as a times b equals b times a. The rational numbers and the real numbers are fields with infinitely many elements.

A finite field has only finitely many elements. The integers modulo a prime p form one, with p elements, which is the most familiar kind. Finite fields are also called Galois fields, after the mathematician Évariste Galois.

## A small example

Take the numbers 0 to 6 and do arithmetic modulo 7, so that every result is replaced by its remainder after dividing by 7.

Then 5 + 4 = 2, because 9 leaves remainder 2 when divided by 7. Also 3 × 5 = 1, so 1/3 = 5 in this field.

## Why proofs use one

Computations are written as arithmetic over the field, so every statement becomes a polynomial equation. Each nonzero number has an inverse, and each result stays among the same fixed set of numbers.

Because the arithmetic is exact, a verifier can test a polynomial at a random point, and a wrong polynomial is very likely to fail that test.

## Related terms

- [Arithmetization](https://redacted.gg/glossary/arithmetization.md): Arithmetization turns a computation into algebra: its steps go into a table, each column becomes a polynomial, and the rules become polynomial equations. STARKs use this form and call it an AIR.
- [Polynomial commitment](https://redacted.gg/glossary/polynomial-commitment.md): A polynomial commitment is a short value that binds a prover to one polynomial. Later the prover can prove the polynomial's value at any chosen point, and the verifier checks that proof against the short value.
- [FRI](https://redacted.gg/glossary/fri.md): FRI is the low-degree test used in STARKs. It checks that committed values come from a polynomial of low degree by halving the problem round after round and testing a few random points, and it rests only on hash functions.
- [STARK](https://redacted.gg/glossary/stark.md): A STARK is a short proof that a long computation was done correctly, which anyone can check quickly. It is built from hash functions alone, so it needs no trusted setup and no elliptic curves, and it can keep the inputs secret.
- [Elliptic curve cryptography](https://redacted.gg/glossary/elliptic-curve.md): Elliptic curve cryptography builds key pairs from points on an elliptic curve over a finite field. It gives about the strength of 3072-bit RSA with 256-bit keys, and Bitcoin signs with the secp256k1 curve.

Sources: [Finite field, Wikipedia](https://en.wikipedia.org/wiki/Finite_field), [The Cambrian explosion of cryptographic proofs, StarkWare](https://starkware.co/integrity-matters-blog/cambrian-explosion-of-cryptographic-proofs/)
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Page: https://redacted.gg/glossary/finite-field/