Elliptic curve method

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^[update], it is still the best algorithm for divisors not greatly exceeding 20 to 25 digits (64 to 83 bits or so), 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 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.

The Lenstra elliptic curve factorization method to find a factor of the given natural number n works as follows:

The time complexity depends on the size of the factor and can be represented by exp((√2 + o(1)) √ln p ln ln p), where p is the smallest factor of n, or ${\displaystyle L_{p}\left[{\frac {1}{2}},{\sqrt {2}}\right]}$, in L-notation.

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