Monday, December 12, 2016

Solutions to Elliptical Curve Problems

Construct the line L through any two points P1 and P2 such that they intersect a third point P3, by direct calculation or using Bezout's theorem:

Soln.

No automatic alt text available.

1) Sketch more of the elliptic curve (2) such that the section is shown for x = 4, y = ?

No automatic alt text available.

The y-coordinate occurs at: y (4)    =  [(4)3     –       (4)  +  1 ] ½     =  [61 ] ½   =  7.8

2)  Use the short Weierstrass form to generate another elliptic curve and graph it. Then obtain the discriminant and ensure it is non-vanishing. Thence obtain h(E).

The short Weierstrass form is:  y2  =   x3     + Ax + B


Let A = -2   and    B = 10   then we will generate:

y2  =   x3    -- 2x + 10

The equation when graphed appears:
No automatic alt text available.
Then the discriminant :  

= -16 (4 A3   +  27 B2 ) =   -16[( 4 (-2)3    + 27(10)2] = 

[ 512  +  (-16)2700 ] =    [512 -  43200]  = -42688




h (E) =   max (4 |A|3 ,   27 B2) =   (4 |-2|3 ,   27 (10)2) =  (32,  2700)

No comments: