# Factoring integers with elliptic curves

```@article{Lenstra1987FactoringIW,
title={Factoring integers with elliptic curves},
author={Hendrik W. Lenstra},
journal={Annals of Mathematics},
year={1987},
volume={126},
pages={649-673}
}```
• H. Lenstra
• Published 1 November 1987
• Mathematics
• Annals of Mathematics
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 25 REFERENCES

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
• 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.
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
• 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"
• J. Pollard
• Computer Science
Mathematical 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.
• Computer Science, Mathematics
STOC '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.
© Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1969, tous droits réservés. L’accès aux archives de la revue « Annales scientifiques de l’É.N.S. » (http://www.
• Computer Science, Mathematics
CACM
• 1983
An encryption method is presented with the novel property that publicly revealing an encryption key does not thereby reveal the corresponding decryption key. This has two important