# 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

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- Mathematics
- 1986

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.

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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…

## References

SHOWING 1-10 OF 37 REFERENCES

Factoring integers with elliptic curves

- Mathematics
- 1986

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.

Factorization of the eighth Fermat number

- Mathematics
- 1981

We describe a Monte Carlo factorization algorithm which was used to factorize the Fermat number F8 = 2256+1. Previously, F8 was known to be composite, but its factors were unknown. Comments Only the…

Theorems on factorization and primality testing

- Computer Science
- 1974

This paper is concerned with the problem of obtaining theoretical estimates for the number of arithmetical operations required to factorize a large integer n or test it for primality, and uses a multi-tape Turing machine for this purpose.

A Monte Carlo factoring algorithm with linear storage

- Computer Science, Mathematics
- 1984

An algorithm which will factor an integer n quite efficiently if the class number h(-n) is free of large prime divisors and the method requires an amount of storage space which is proportional to the length of the input n.

A method of factoring and the factorization of

- Mathematics
- 1975

The continued fraction method for factoring integers, which was introduced by D. H. Lehmer and R. E. Powers, is discussed along with its computer implementation. The power of the method is…

An improved Monte Carlo factorization algorithm

- Computer Science
- 1980

A cycle-finding algorithm is described which is about 36 percent faster than Floyd's (on the average), and applied to give a Monte Carlo factorization algorithm which is similar to Pollard's but about 24 percent faster.

Modular multiplication without trial division

- Mathematics, Computer Science
- 1985

A method for multiplying two integers modulo N while avoiding division by N, a representation of residue classes so as to speed modular multiplication without affecting the modular addition and subtraction algorithms.

The multiple polynomial quadratic sieve

- Computer Science
- 1987

A modification, due to Peter Montgomery, of Pomerance's Quadratic Sieve for factoring large integers is discussed along with its implementation, which enables one to factor numbers in the 60-digit range in about a day, using a large minicomputer.

Sequences of numbers generated by addition in formal groups and new primality and factorization tests

- Mathematics
- 1986

A $p+1$ method of factoring

- Mathematics
- 1982

Let N have a prime divisor p such that p + 1 has only small prime divisors. A method is described which will allow for the determination of p, given N. This method is analogous to the p — 1 method of…