The block Wiedemann algorithm for computing kernel vectors of a matrix over a finite field is a generalization by Don Coppersmith of an algorithm due to Doug Wiedemann.
square matrix over some finite field F, let
be a random vector of length
Consider the sequence of vectors
obtained by repeatedly multiplying the vector by the matrix
be any other vector of length
, and consider the sequence of finite-field elements
has a minimal polynomial; by the Cayley–Hamilton theorem we know that this polynomial is of degree (which we will call
; so the minimal polynomial of the matrix annihilates the sequence
But the Berlekamp–Massey algorithm allows us to calculate relatively efficiently some sequence
Our hope is that this sequence, which by construction annihilates
We then take advantage of the initial definition of
is a hopefully non-zero kernel vector of
The natural implementation of sparse matrix arithmetic on a computer makes it easy to compute the sequence S in parallel for a number of vectors equal to the width of a machine word – indeed, it will normally take no longer to compute for that many vectors than for one.
If you have several processors, you can compute the sequence S for a different set of random vectors in parallel on all the computers.
It turns out, by a generalization of the Berlekamp–Massey algorithm to provide a sequence of small matrices, that you can take the sequence produced for a large number of vectors and generate a kernel vector of the original large matrix.
max
max
max
max
max
max
max
max
are a series of vectors of length n; but in practice you can take
as a sequence of unit vectors and simply write out the first
entries in your vectors at each time t. The block Wiedemann algorithm can be used to calculate the leading invariant factors of the matrix, ie, the largest blocks of the Frobenius normal form.
is a finite field of size
invariant factors of
{\displaystyle p\geq {\begin{cases}1/64,&{\text{if }}b=k+1{\text{ and }}q=2\\\left(1-{\frac {3}{2^{b-k}}}\right)^{2}\geq 1/16&{\text{if }}b\geq k+2{\text{ and }}q=2\\\left(1-{\frac {2}{q^{b-k}}}\right)^{2}\geq 1/9&{\text{if }}b\geq k+1{\text{ and }}q>2\end{cases}}}