# Block Lanczos algorithm for nullspace of a matrix over a finite field

The

**Block Lanczos algorithm for nullspace of a matrix over a finite field**is a procedure for finding thenullspace 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 [*cite journal |last= Montgomery|first= P L |year=1995 |title=A Block Lanczos Algorithm for Finding Dependencies over GF(2)|journal=EUROCRYPT '95 |volume=921 |pages=106–120*] ; it is based on, and bears a strong resemblance to, theLanczos algorithm for findingeigenvalue s 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 factorisation algorithms such as thequadratic sieve andnumber 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 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.**References**

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**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… … Wikipedia**List of mathematics articles (B)**— NOTOC B B spline B* algebra B* search algorithm B,C,K,W system BA model Ba space Babuška Lax Milgram theorem Baby Monster group Baby step giant step Babylonian mathematics Babylonian numerals Bach tensor Bach s algorithm Bachmann–Howard ordinal… … Wikipedia**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.Montgomery is particularly… … Wikipedia