Isogenies - Elliptic Curves: Number Theory and Cryptography - page 402

12.4 Let E 1 : y 1 = x 1 + ax 1 + bx 1 be an elliptic curve over some field of
characteristic not 2 with b =0and a 2 − 4 b =0. Let E 2 be the elliptic

curve y 2 = x 2 − 2 ax 2 +( a 2 4 b ) x 2 . Define α by

( x 2 ,y 2 )= α ( x 1 ,y 1 )= y 1

−

.

y 1 ( x 1 − b )
x 1 x 1 ,

Let s i =1 /y i and t i = x i /y i .Then t i and s i are 0 at ∞ (in fact, t i has a

simple zero at ∞ and s i has a triple zero at ∞ , but we won't use this).

We want to show that α ( . To do this, whenever we encounter

an expression 0 / 0or ∞/∞ , we rewrite it so as to obtain an expression
in which every part is defined.

∞ )= ∞

(a) Show that

s 1

2 = s 1

1
1 − b ( s 1 /t 1 ) 2 .

s 2 =

1 − b ( s 1 /t 1 ) 2 , t 1

(b) Show that s 1 /t 1 = t 1 + as 1 t 1 + bs 1 ,so s 1 /t 1 is 0 at ∞ .
(c) Write

= t 1 + as 1 + b s 1

2

s 1
t 1

t 1 .

t 1

Show that s 1 /t 1 has the value 0 at ∞ .

(d) Show that α maps ∞ on E 1 to ∞ on E 2 .

12.5 Let E 1 ,E 2 ,α,s 2 ,t 2 be as in Exercise 12.4.

(a) Show that

x 1 y 1 y 1

s 2 =

( x 1 + ax 1 + b )( x 1 − b ) ,

2 =

x 1 − b .

(b) Conclude that α (0 , 0) = ∞ .

12.6 Let E be an elliptic curve given by a generalized Weierstrass equation
y 2 + a 1 xy + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6 .Let P =( x P ,y P )and Q =
( x Q ,y Q )bepointson E .Let x P + Q ,y P + Q denote the x and y coordinates
of the point P + Q .

(a) Show that if 2 Q = ∞ ,then u Q =0and

v Q
x P −

a 1 ( x P − x Q )+ y P − y Q

x P + Q −x Q =

x Q ,

P + Q −y Q = − v Q .

( x P − x Q ) 2