Numerical Analysis over Finite Fields

Question

Notwithstanding that it isn't numerical analysis if it's over finite fields, but what topics that are traditionally considered part of numerical analysis still have some substance to them if the reals are replaced with finite fields or an algebraic closure thereof? Perhaps using Hamming distance as a metric for convergence purposes, with convergence of an iteration in a discrete setting just meaning that the Hamming distance between successive iterations becomes zero i.e. the algorithm has a fixed-point.

I ask about still having substance because I suspect that in the ff setting, na topics will mostly either not make sense, or be trivial.

Answer 1

There is substantial interest in vector spaces over finite fields. Much of the research falls into the area of Computational group theory or computations in Modular Representation theory. In particular Gauss-Jordan Elimination works and if I recall correctly has the same computational complexity as over characteristic 0.

Answer 2

The people who factor large numbers using sieve algorithms (the quadratic sieve, the special and general number field sieves) wind up with enormous (millions by millions) systems of linear equations over the field of two elements, and they need to put a lot of thought into the most efficient ways to solve these systems if they want their algorithms to terminate before the sun does.