Content deleted Content added
m →External links: HTTP to HTTPS for SourceForge |
m Fixed grammar Tags: canned edit summary Mobile edit Mobile app edit iOS app edit App select source |
||
Line 9:
The Lenstra elliptic-curve factorization method to find a factor of a given natural number <math>n</math> works as follows:
# Pick a random [[elliptic curve]] over <math>\mathbb{Z}/n\mathbb{Z}</math> (the integers modulo <math>n</math>), with equation of the form <math>y^2 = x^3 + ax + b \pmod n</math> together with a non-trivial [[Point (geometry)|point]] <math>P(x_0,y_0)</math> on it.
#:This can be done by first picking random <math>x_0,y_0,a \in \mathbb{Z}/n\mathbb{Z}</math>, and then setting <math>b = y_0^2 - x_0^3 - ax_0\pmod n</math> to
# One can define ''Addition'' of two points on the curve, to define a [[Group (mathematics)|group]]. The addition laws are given in the [[elliptic curve#The group law|article on elliptic curves]].
#:We can form repeated multiples of a point <math>P</math>: <math>[k]P = P + \ldots + P \text{ (k times)}</math>. The addition formulae involve taking the modular slope of a chord joining <math>P</math> and <math>Q</math>, and thus division between residue classes modulo <math>n</math>, performed using the [[extended Euclidean algorithm]]. In particular, division by some <math>v \bmod n</math> includes calculation of the <math>\gcd(v,n)</math>.
| |||