Strategies in filtering in the number field sieve

A critical step when factoring large integers by the Number Field Sieve consists of finding dependencies in a huge sparse matrix over the field GF(2), using a Block Lanczos algorithm. Both size and weight (the number of non-zero elements) of the matrix critically affect the running time of Block Lanczos. In order to keep size and weight small the relations coming out of the siever do not flow directly into the matrix, but are filtered first in order to reduce the matrix size. This paper discusses several possible filter strategies and their use in the recent record factorizations of RSA-140, R211 and RSA-155.