The Basic Theory - Elliptic Curves: Number Theory and Cryptography - page 72

(b) Note that P Q (mod 5). Compute 2 P on E mod 5. Show that

the answer is the same as ( P + Q ) mod 5. Observe that since P

≡

Q ,
the formula for adding the points mod 5 is not the reduction of the
formula for adding P + Q . However, the answers are the same. This
shows that the fact that reduction mod a prime is a homomorphism
is subtle, and this is the reason for the complicated formulas in
Section 2.11.

≡

2.5 Let ( x, y ) be a point on the elliptic curve E given by y 2 = x 3 + Ax + B .
Show that if y =0then3 x 2 + A =0. ( Hint: What is the condition for
a polynomial to have x as a multiple root?)

2.6 Show that three points on an elliptic curve add to ∞ ifandonlyifthey
are collinear.
2.7 Let C be the curve u 2 + v 2 = c 2 1+ du 2 v 2 , as in Section 2.6.3. Show
that the point ( c, 0) has order 4.

2.8 Show that the method at the end of Section 2.2 actually computes kP .
( Hint : Use induction on the length of the binary expansion of k . f
k = k 0 +2 k 1 +4 k 2 + ··· +2 a , assume the result holds for k = k 0 +
2 k 1 +4 k 2 + ··· +2 − 1 a − 1 .)

2.9 If P =( x, y ) = ∞ is on the curve described by (2.1), then −P is the
other finite point of intersection of the curve and the vertical line through
P . Show that −P =( x, −a 1 x − a 3 − y ). ( Hint: This involves solving
a quadratic in y . Note that the sum of the roots of a monic quadratic
polynomial equals the negative of the coe cient of the linear term.)

2.10 Let R be the real numbers. Show that the map ( x, y, z ) ( x : y : z )

gives a two-to-one map from the sphere x 2 + y 2 + z 2 =1in R 3 to P 2

→

.

R

Since the sphere is compact, this shows that P 2 is compact under the

topology inherited from the sphere (a set is open in P 2

R

if and only if

R

its inverse image is open in the sphere).

2.11 (a) Show that two lines a 1 x + b 1 y + c 1 z =0and a 2 x + b 2 y + c 2 z =0
in two-dimensional projective space have a point of intersection.
(b) Show that there is exactly one line through two distinct given points
in P 2 K .
2.12 Suppose that the matrix

⎛ ⎞

a 1 b 1
a 2 b 2
a 3 b 3

⎝ ⎠

M =

has rank 2. Let ( a, b, c ) be a nonzero vector in the left nullspace of M ,
so ( a, b, c ) M = 0. Show that the parametric equations

x = a 1 u + b 1 v, y = a 2 u + b 2 v, z = a 3 u + b 3 v,