Speeding the Pollard and elliptic curve methods of factorization

Since 1974, several algorithms have been developed that attempt to factor a large number N by doing extensive computations modulo 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 $p \pm 1$ and ECM, looking for a match. Polynomial preconditioning lets us search a fixed table of size n with $n/2 + o(n)$ multiplications. A parametrization of elliptic curves lets Step 1 of ECM compute the x-coordinate of nP from that of P in about 9.3 ${\log _2}$ n multiplications for arbitrary P.