One of the last steps of GNFS is to solve a large matrix-vector equation (usually using the Block Lanczos algorithm or the Block Wiedemann algorithm). The matrix for the most recent RSA number factored (RSA-220, completed in May 2016) was an $M\times M$ square where $M=192796550$, making this a cumbersome part of the algorithm. Can we estimate how big $M$ will be for some number $N$ that we wish to factor? Is there an asymptotic relation between $M$ and $N$ in $\mathcal{O}(\cdot )$ notation?
2026-02-22 23:42:41.1771803761
In the General Number Field Sieve, can we estimate the size of the matrix in terms of the number being factored?
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