Cryptography Reference
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8.4 Let E be given by y 2 = x 3 + Ax + B with A, B
Z .Let P =( x, y )be
a point on E .
(a) Let 2 P =( x 2 ,y 2 ). Show that
y 2 4 x 2 (3 x 2 +4 A )
3 x 2 +5 Ax +27 B =4 A 3 +27 B 2 .
(b) Show that if both P and 2 P have coordinates in Z ,then y 2 divides
4 A 3 +27 B 2 . This gives another way to finish the proof of the
Lutz-Nagell theorem.
8.5 Let E be the elliptic curve over Q given by y 2 + xy = x 3 + x 2
11 x .
Show that the point
P = 11
11
8
4 , −
is a point of order 2. This shows that the integrality part of Theorem 8.7
(see also Exercise 8.2), which is stated for Weierstrass equations, does
not hold for generalized Weierstrass equations. However, since changing
from generalized Weierstrass form to the form in Exercise 8.2 affects
only powers of 2 in the denominators, only the prime 2 can occur in the
denominators of torsion points in generalized Weierstrass form.
8.6 Show that the Mordell-Weil group E ( Q ) of the elliptic curve y 2 = x 3
−x
is isomorphic to Z 2 Z 2 .
8.7 Suppose E ( Q ) is generated by one point Q of infinite order. Suppose
we take R 1 =3 Q , which generates E ( Q ) / 2 E ( Q ). Show that the process
with P 0 = Q and
P i = R j i +2 P i +1 ,
as in Section 8.3, never terminates. This shows that a set of represen-
tatives of E ( Q ) / 2 E ( Q ) does not necessarily generate E ( Q ).
8.8 Show that there is a set of representatives of E ( Q ) / 2 E ( Q ) that gener-
ates E ( Q ). ( Hint: This mostly follows from the Mordell-Weil theorem.
However, it does not handle the odd order torsion. Use Corollary 3.13
to show that the odd order torsion in E ( Q ) is cyclic. In the set of rep-
resentatives, use a generator of this cyclic group for the representative
of the trivial coset.)
8.9 Let E be an elliptic curve defined over Q and let n be a posi tiv e integer.
Assume that E [ n ]
E ( Q ). Let P
E ( Q )andlet Q
E ( Q )besuch
that nQ = P . Define a map δ P : Gal( Q / Q )
E [ n ]by δ P ( σ )= σQ
Q .
(a) Let σ ∈ Gal( Q / Q ). Show that σQ − Q ∈ E [ n ].
(b) Show that δ P is a cocycle in Z ( G, E [ n ]).
 
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