# Speeding the Pollard and elliptic curve methods of factorization

```@article{Montgomery1987SpeedingTP,
title={Speeding the Pollard and elliptic curve methods of factorization},
author={Peter L. Montgomery},
journal={Mathematics of Computation},
year={1987},
volume={48},
pages={243-264}
}```
Since 1974, several algorithms have been developed that attempt to factor a large number N by doing extensive computations module N and occasionally taking GCDs with N. These began with Pollard's p 1 and Monte Carlo methods. More recently, Williams published a p + 1 method, and Lenstra discovered an elliptic curve method (ECM). We present ways to speed all of these. One improvement uses two tables during the second phases of p ? 1 and ECM, looking for a match. Polynomial preconditioning lets us…
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