About: Lenstra elliptic-curve factorization

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The Lenstra elliptic-curve factorization or the elliptic-curve factorization method (ECM) is a fast, sub-exponential running time, algorithm for integer factorization, which employs elliptic curves. For general-purpose factoring, ECM is the third-fastest known factoring method. The second-fastest is the multiple polynomial quadratic sieve, and the fastest is the general number field sieve. The Lenstra elliptic-curve factorization is named after Hendrik Lenstra.

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rdfs:label	Lenstra elliptic-curve factorization (en)
rdfs:comment	The Lenstra elliptic-curve factorization or the elliptic-curve factorization method (ECM) is a fast, sub-exponential running time, algorithm for integer factorization, which employs elliptic curves. For general-purpose factoring, ECM is the third-fastest known factoring method. The second-fastest is the multiple polynomial quadratic sieve, and the fastest is the general number field sieve. The Lenstra elliptic-curve factorization is named after Hendrik Lenstra. (en)
dbp:wikiPageUsesTemplate	dbt:As_of dbt:Cite_book dbt:Cite_journal dbt:Harvtxt dbt:Main dbt:Math dbt:Mvar dbt:Reflist dbt:Sqrt dbt:Technical dbt:Number_theoretic_algorithms
Subject	Finite fields Integer factorization algorithms
prov:wasDerivedFrom	http://en.wikipedia.org/wiki/Lenstra_elliptic-curve_factorization?oldid=1068199484&ns=0
Wikipage page ID	154212 (xsd:integer)
page length (characters) of wiki page	26754 (xsd:nonNegativeInteger)
Wikipage revision ID	1068199484 (xsd:integer)
Link from a Wikipage to another Wikipage	Greatest common divisor Group (mathematics) Nadia Heninger Projective space Smooth number Annals of Mathematics Montgomery curve General number field sieve Finite fields Integer factorization algorithms Divisor Quadratic sieve Daniel J. Bernstein Decimal Hyperelliptic curve Linearity Computer Coprime integers EXPTIME Edwards curve Hendrik Lenstra Heuristic Notices of the American Mathematical Society Elliptic-curve cryptography Elliptic curve point multiplication Finite field Extended Euclidean algorithm Factorial Fermat's little theorem Grover's algorithm Hasse's theorem on elliptic curves Integer factorization L-notation Lagrange's theorem (group theory) Modular arithmetic Modular multiplicative inverse UBASIC Weierstrass elliptic function Euclidean algorithm Strong prime Torsion group Mathematics of Computation Multiplicative group Point (geometry) Pollard's p − 1 algorithm dbr:Canfield–Erdős–Pomerance_theorem
has abstract	The Lenstra elliptic-curve factorization or the elliptic-curve factorization method (ECM) is a fast, sub-exponential running time, algorithm for integer factorization, which employs elliptic curves. For general-purpose factoring, ECM is the third-fastest known factoring method. The second-fastest is the multiple polynomial quadratic sieve, and the fastest is the general number field sieve. The Lenstra elliptic-curve factorization is named after Hendrik Lenstra. Practically speaking, ECM is considered a special-purpose factoring algorithm, as it is most suitable for finding small factors. Currently, it is still the best algorithm for divisors not exceeding 50 to 60 digits, as its running time is dominated by the size of the smallest factor p rather than by the size of the number n to be factored. Frequently, ECM is used to remove small factors from a very large integer with many factors; if the remaining integer is still composite, then it has only large factors and is factored using general-purpose techniques. The largest factor found using ECM so far has 83 decimal digits and was discovered on 7 September 2013 by R. Propper. Increasing the number of curves tested improves the chances of finding a factor, but they are not linear with the increase in the number of digits. (en)
foaf:isPrimaryTopicOf	http://en.wikipedia.org/wiki/Lenstra_elliptic-curve_factorization
is Wikipage redirect of	Lenstra Elliptic Curve Factorization Lenstra elliptic curve factorization Elliptic curve method Elliptic Curve Factorization Method Elliptic curve factorisation Elliptic curve factorization Elliptic curve factorization method Lenstra's ECM
is Link from a Wikipage to another Wikipage of	Magma (computer algebra system) Algebraic-group factorisation algorithm Arjen Lenstra Lenstra Elliptic Curve Factorization Lenstra elliptic curve factorization List of distributed computing projects List of number theory topics ECM Hendrik Lenstra Hessian form of an elliptic curve List of algorithms Elliptic-curve cryptography Elliptic-curve cryptography Elliptic curve method Elliptic curve primality Fermat number Integer factorization Euclidean algorithm Multiplicative group of integers modulo n Elliptic Curve Factorization Method Elliptic curve factorisation Elliptic curve factorization Elliptic curve factorization method Lenstra's ECM
is foaf:primaryTopic of	http://en.wikipedia.org/wiki/Lenstra_elliptic-curve_factorization

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