What is a finite field?

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.

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.

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.