# 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} }

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…

## 1,109 Citations

### Integer Factorization of Semi-Primes Based on Analysis of a Sequence of Modular Elliptic Equations

- Mathematics, Computer ScienceInt. J. Commun. Netw. Syst. Sci.
- 2011

A non-deterministic algorithm that computes a factor of a semi-prime integer n=pq, where prime factors p and q are unknown is provided.

### On the number of elliptic pseudoprimes

- Mathematics
- 1989

For an elliptic curve E with complex multiplication by an order in K = Q(V\/d), a point P of infinite order on E, and any prime p with (-d I p) = -1, we have that (p + 1) * P = O(mod p), where 0 is…

### Factoring integers with the number field sieve

- Mathematics
- 1993

In 1990, the ninth Fermat number was factored into primes by means of a new algorithm, the “number field sieve”, which was proposed by John Pollard. The present paper is devoted to the description…

### Estimates related to the arithmetic of elliptic curves

- Mathematics
- 2007

This dissertation presents results related to two problems in the arithmetic of elliptic curves. Let En : y 2 = x − nx denote the family of congruent number elliptic curves. In [13], Feng and Xiong…

### Fe b 20 09 The distribution of the number of points modulo an integer on elliptic curves over finite fields

- Mathematics
- 2009

Let Fq be a finite field and let b and N be integers. We study the probability that the number of points on a randomly chosen elliptic curve E over Fq equals b modulo N . We prove explicit formulas…

### Discrete logarithms and local units

- MathematicsPhilosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences
- 1993

Let K be a number field and (9K its ring of integers. Let l be a prime number and e a positive integer. We give a method to construct leth powers in (9K using smooth algebraic integers. This method…

### A rigorous time bound for factoring integers

- Mathematics
- 1992

In this paper a probabilistic algorithm is exhibited that factors any positive integer n into prime factors in expected time at most Ln[2, 1 + o()] for n oo, where L,[a, b] = exp(b(logx)a(loglogx)l…

### Constructing elliptic curves of prime order

- Mathematics, Computer Science
- 2007

A very efficient algorithm to construct an elliptic curve E and a finite field F such that the order of the point group E(F) is a given prime number N, so fast that it may be used to tackle the related problem of finding elliptic curves with point groups of prime order of prescribed size.

### O ct 2 01 2 AVERAGE FROBENIUS DISTRIBUTION FOR ELLIPTIC CURVES DEFINED OVER FINITE GALOIS EXTENSIONS OF THE RATIONALS ( APPEARED IN MATHEMATICAL PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY )

- Mathematics
- 2014

Abstract. Let K be a fixed number field, assumed to be Galois over Q. Let r and f be fixed integers with f positive. Given an elliptic curve E, defined over K, we consider the problem of counting the…

### HOW TO FIND SMOOTH PARTS OF INTEGERS

- Mathematics
- 2004

Let P be a finite set of primes, and let S be a finite sequence of positive integers. This paper presents an algorithm to find the largest P smooth divisor of each integer in S. The algorithm takes…

## References

SHOWING 1-10 OF 25 REFERENCES

### 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.

### 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.

### Abelian varieties over finite fields

- Mathematics
- 1969

© 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.…

### A method for obtaining digital signatures and public-key cryptosystems

- Computer Science, MathematicsCACM
- 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…