Summary at: [1] http://rjlipton.wordpress.com/2012/08/09/a-new-way-to-solve-linear-equations/

Original paper by Prasad Raghavendra at: [2] http://www.eecs.berkeley.edu/~prasad/linsystems.pdf

This is to solve a linear equation system \(Ax=b\) over a finite field \(\mathbb{F}_q\). The finite field is has a finite size of \(q\) (a prime), we can think of it as integers 0 to \(q-1\), with the addition and multiplication are done in modulus \(q\). Matrix \(A\) is of order \(m\times n\).

The algorithm to solve the problem is as follows: It first takes \(N=Cn\) random vectors from \(\mathbb{F}|n_q\), make it into a set \(S_0\). Then filter \(S_0\) to find those who can satisfy the first row of \(Ax=b\). The satisfying random vectors are then “recombined”: Select \(q-1\) of them randomly and sum up to give a new vector. Repeat this random select and sum up process to produce \(N\) vectors. These \(N\) vectors are make into a set \(S_1\), then applied to the second row of \(Ax=b\). The filtered set at the last row are the solutions for the linear equation.

Note that, in the finite field, there could be infinitely many solutions for a linear system. Therefore a set of solutions is produced. The original paper [2] gives the analysis on the size of \(N\) required and the performance bound.

## Bibliographic data

```
@misc{
title = "A Randomized Algorithm for Linear Equations over Prime Fields",
author = "Prasad Raghavendra",
month = "Aug",
year = "2012",
}
```