Cryptography Reference
In-Depth Information
describe the line ax + by + cz =0in P 2 K . (It is easy to see that the
points ( x : y : z ) lie on the line. The main point is that each point on
the line corresponds to a pair ( u, v ).)
(a) Put the Legendre equation y 2 = x ( x − 1)( x − λ ) into Weierstrass
form and use this to show that the j -invariant is
2.13
j =2 8 ( λ 2
− λ +1) 3
λ 2 ( λ − 1) 2
.
(b) Show that if j
=0 , 1728 then there are six distinct values of λ
giving this j ,andthatif λ is one such value then the full set is
{λ, 1
1
λ
1 , λ
1
λ , 1 − λ,
λ ,
}.
1
λ
λ
(c) Show that if j = 1728 then λ = 1 , 2 , 1 / 2, and if j =0then
λ 2
− λ +1=0.
2.14 Consider the equation u 2
− v 2 = 1, and the point ( u 0 ,v 0 )=(1 , 0).
(a) Use the method of Section 2.5.4 to obtain the parameterization
u = m 2 +1
m 2
2 m
m 2
1 ,
v =
1 .
(b) Show that the projective curve u 2
− v 2
= w 2
has two points at
infinity, (1 : 1 : 0) and (1 : 1:0).
(c) The parameterization obtained in (a) can be written in projective
coordinates as ( u : v : w )=( m 2 +1:2 m : m 2
1) (or ( m 2 + n 2 :
2 mn : m 2
− n 2 ) in a homogeneous form). Show that the values
m = ± 1 correspond to the two points at infinity. Explain why this
is to be expected from the graph (using real numbers) of u 2
−v 2 =1.
( Hint: Where does an asymptote intersect a hyperbola?)
2.15 Suppose ( u 0 ,v 0 ,w 0 )=( u 0 , 0 , 0) lies in the intersection
au 2 + bv 2 = e,
cu 2 + dw 2 = f.
(a) Show that the procedure of Section 2.5.4 leads to an equation of
the form “square = degree 2 polynomial in m .”
(b) Let F = au 2 + bv 2
= e and G = cu 2 + dw 2
= f . Show that the
Jacobian matrix F u F v F w
at ( u 0 , 0 , 0) has rank 1. Since the
G u G v G w
rank is less than 2, this means that the point is a singular point.
2.16 Show that the cubic equation x 3 + y 3
= d can be transformed to the
elliptic curve y 1 = x 1 432 d 2 .
 
Search WWH ::




Custom Search