# 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…
1,214 Citations
Factoring integers with elliptic curves
This paper is devoted to the description and analysis of a new algorithm to factor positive integers that depends on the use of elliptic curves and it is conjectured that the algorithm determines a non-trivial divisor of a composite number n in expected time at most K( p)(log n)2.
The Elliptic Curve Method for Factoring Paul Zimmermann , INRIA
The Elliptic Curve Method can be viewed as a generalization of Pollard's p − 1 method, just like ECPP generalizes the n − 1 primality test, which relies on Hasse’s theorem.
Speeding up Integer Multiplication and Factorization
Improvements to well-known algorithms for integer multiplication and factorization are explored, showing how parameters for these algorithms can be chosen accurately, taking into account the distribution of prime factors in integers produced by NFS to obtain an accurate estimate of finding a prime factor with given parameters.
The factorization of the ninth Fermat number
• Mathematics
• 1993
In this paper we exhibit the full prime factorization of the ninth Fermat number F9 = 2(512) + 1. It is the product of three prime factors that have 7, 49, and 99 decimal digits. We found the two
Deterministic elliptic curve primality proving for a special sequence of numbers
• Mathematics, Computer Science
• 2013
We give a deterministic algorithm that very quickly proves the primality or compositeness of the integers N in a certain sequence, using an elliptic curve E/Q with complex multiplication by the ring
Speeding up elliptic curve discrete logarithm computations with point halving
• Computer Science, Mathematics
Des. Codes Cryptogr.
• 2013
A careful analysis of the alternative rho method with new iteration function is presented, and generally the new method can achieve a significant speedup for computing elliptic curve discrete logarithms over binary fields.
Counting Points On Elliptic Curves Over F p n Using Couveignes's Algorithm
• Computer Science, Mathematics
• 1996
The aim of this paper is to describe the rst successful implementation of Couveignes's algorithm and to give numerous computational examples, including the use of fast algorithms for performing incremental operations on series.
Primality Proving Using Elliptic Curves: An Update
• F. Morain
• Mathematics, Computer Science
ANTS
• 1998
An account of the recent theoretical and practical improvements of ECPP, as well as new benchmarks for integers of various sizes and a new primality record are given.
Addendum: The Factorization of the Ninth Fermat Number
• Mathematics
• 1995
In this paper we exhibit the full prime factorization of the ninth Fermat number Fg = 2512 + 1 . It is the product of three prime factors that have 7, 49, and 99 decimal digits. We found the two