# Factoring integers with elliptic curves

@article{Lenstra1986FactoringIW, title={Factoring integers with elliptic curves}, author={Hendrik W. Lenstra}, journal={Annals of Mathematics}, year={1986}, volume={126}, pages={649-673} }

This paper is devoted to the description and analysis of a new algorithm to factor positive integers. It depends on the use of elliptic curves. The new method is obtained from Pollard's (p - 1)-method (Proc. Cambridge Philos. Soc. 76 (1974), 521-528) by replacing the multiplicative group by the group of points on a random elliptic curve. 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, where p is the least…

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## References

SHOWING 1-10 OF 32 REFERENCES

### Elliptic Curves Over Finite Fields and the Computation of Square Roots mod p

- Mathematics, Computer Science
- 1985

A deterministic algorithm to compute the number of F^-points of an elliptic curve that is defined over a finite field Fv and which is given by a Weierstrass equation is presented.

### Speeding the Pollard and elliptic curve methods of factorization

- Mathematics
- 1987

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…

### 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 probabilistic factorization algorithm with quadratic forms of negative discriminant

- Computer Science, Mathematics
- 1987

We propose a probabilistic algorithm for factorization of an integer N with run time (expVlog N log log N)V54 +?(1). Asymptotically, our algorithm will be as fast as the wellknown factorization…

### Arithmetic moduli of elliptic curves

- Mathematics
- 1985

This work is a comprehensive treatment of recent developments in the study of elliptic curves and their moduli spaces. The arithmetic study of the moduli spaces began with Jacobi's "Fundamenta Nova"…

### Theorems on factorization and primality testing

- Computer ScienceMathematical Proceedings of the Cambridge Philosophical Society
- 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.

### Prime numbers and computer methods for factorization

- Mathematics
- 1985

1. The Number of Primes Below a Given Limit.- What Is a Prime Number?.- The Fundamental Theorem of Arithmetic.- Which Numbers Are Primes? The Sieve of Eratosthenes.- General Remarks Concerning…

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

- Mathematics
- 1986

### Almost all primes can be quickly certified

- Computer Science, MathematicsSTOC '86
- 1986

A new probabilistie primality test is presented, different from the tests of Miller, Solovay-Strassen, and Rabin in that its assertions of primality are certain, rather than being correct with high probability or dependent on an unproven assumption.