Editing Lenstra elliptic-curve factorization (section)

==Stage 2==

The above text is about the first stage of elliptic curve factorisation. There one hopes to find a prime divisor {{mvar|p}} such that <math>sP</math> is the neutral element of <math>E(\mathbb{Z}/p\mathbb{Z})</math>.
In the second stage one hopes to have found a prime divisor {{mvar|q}} such that <math>sP</math> has small prime order in <math>E(\mathbb{Z}/q\mathbb{Z})</math>.

We hope the order to be between <math>B_1</math> and <math>B_2</math>, where <math>B_1</math> is determined in stage 1 and <math>B_2</math> is new stage 2 parameter.
Checking for a small order of <math>sP</math>, can be done by computing <math>(ls)P</math> modulo {{mvar|n}} for each prime {{mvar|l}}.