Block Lanczos algorithm for nullspace of a matrix over a finite field
Encyclopedia
The block Lanczos algorithm for nullspace of a matrix over a finite field is a procedure for finding the nullspace of a matrix
using only multiplication of the matrix by long, thin matrices. These long, thin matrices are considered as vectors of tuples of finite-field entries, and so tend to be called 'vectors' in descriptions of the algorithm.
It was developed by Peter Montgomery
and published in 1995; it is based on, and bears a strong resemblance to, the Lanczos algorithm
for finding eigenvalues of large sparse real matrices.
The Block Lanczos algorithm is amongst the most efficient methods known for finding nullspaces, which is the final stage in integer factorization
algorithms such as the quadratic sieve
and number field sieve, and its development has been entirely driven by this application.
The block Wiedemann algorithm
is more useful in contexts where several systems each large enough to hold the entire matrix are available, since in that algorithm the systems can run independently until a final stage at the end.
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
using only multiplication of the matrix by long, thin matrices. These long, thin matrices are considered as vectors of tuples of finite-field entries, and so tend to be called 'vectors' in descriptions of the algorithm.
It was developed by Peter Montgomery
Peter Montgomery
Peter Lawrence Montgomery is an American mathematician who has published widely in the more mathematical end of the field of cryptography. He is currently a researcher in the cryptography group at Microsoft Research....
and published in 1995; it is based on, and bears a strong resemblance to, the Lanczos algorithm
Lanczos algorithm
The Lanczos algorithm is an iterative algorithm invented by Cornelius Lanczos that is an adaptation of power methods to find eigenvalues and eigenvectors of a square matrix or the singular value decomposition of a rectangular matrix. It is particularly useful for finding decompositions of very...
for finding eigenvalues of large sparse real matrices.
The Block Lanczos algorithm is amongst the most efficient methods known for finding nullspaces, which is the final stage in integer factorization
Integer factorization
In number theory, integer factorization or prime factorization is the decomposition of a composite number into smaller non-trivial divisors, which when multiplied together equal the original integer....
algorithms such as the quadratic sieve
Quadratic sieve
The quadratic sieve algorithm is a modern integer factorization algorithm and, in practice, the second fastest method known . It is still the fastest for integers under 100 decimal digits or so, and is considerably simpler than the number field sieve...
and number field sieve, and its development has been entirely driven by this application.
Parallelisation issues
The algorithm is essentially not parallel: it is of course possible to distribute the matrix-'vector' multiplication, but the whole vector must be available for the combination step at the end of each iteration, so all the machines involved in the calculation must be on the same fast network. In particular, it is not possible to widen the vectors and distribute slices of vectors to different independent machines.The block Wiedemann algorithm
Block Wiedemann algorithm
The block Wiedemann algorithm for computing kernel vectors of a matrix over a finite field is a generalisation of an algorithm due to Don Coppersmith.- Coppersmith's algorithm :...
is more useful in contexts where several systems each large enough to hold the entire matrix are available, since in that algorithm the systems can run independently until a final stage at the end.