Speeding the Pollard and Elliptic Curve Methods of Factorization

Peter L. Montgomery
1987 Mathematics of Computation  
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 ± 1 and ECM, looking for a match. Polynomial preconditioning lets
more » ... 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 log2 n multiplications for arbitrary P.
doi:10.2307/2007888 fatcat:6zfknzpvu5acflxq5gfz6vmkiy