The Basic Theory - Elliptic Curves: Number Theory and Cryptography - page 56

LEMMA 2.24
Let E be the ellipticcurve y 2 = x 3 + Ax + B .Fixapoint ( u, v ) on E .W rite

( x, y )+( u, v )=( f ( x, y ) ,g ( x, y )) ,

where f ( x, y ) and g ( x, y ) are rationalfunctions of x, y (the coe cients depend
on ( u, v ) )and y is regarded as a function of x satisfying dy/dx =(3 x 2 +
A ) / (2 y ) .Then

d

dx f ( x, y )
g ( x, y )

1
y .

=

PROOF The addition formulas give
f ( x, y )= y − v
x − u

2

− x − u

v ) 3 + x ( y u ) 2 +2 u ( y u ) 2 u ) 3

g ( x, y )= −

( y − − v )( x − − v )( x − − v ( x −

( x − u ) 3

dx f ( x, y )= 2 y ( y − v )( x − u ) − 2( y − v ) 2

− ( x − u ) 3 d

.

( x − u ) 3

A straightforward but lengthy calculation, using the fact that 2 yy =3 x 2 + A ,
yields

d
dx f ( x, y ) − g ( x, y ))

( x − u ) 3 ( y

= v ( Au + u 3

− v 2 − Ax − x 3 + y 2 )+ y ( −Au − u 3 + v 2 + Ax + x 3 − y 2 ) .

Since ( u, v )and( x, y )areon E ,wehave v 2 = u 3 + Au + B and y 2 = x 3 + Ax + B .
Therefore, the above expression becomes

v ( − B + B )+ y ( B − B )=0 .

Therefore, y dx f ( x, y )= g ( x, y ).

REMARK 2.25 Lemma 2.24 is perhaps better stated in terms of differ-
entials. It says that the differential dx/y is translation invariant. In fact, it
is the unique translation invariant differential, up to scalar multiples, for E .
See [109].

LEMMA 2.26
Let α 1 ,α 2 ,α 3 be nonzero endom orphism s of an ellipticcurve E with α 1 + α 2 =
α 3 .W rite

α j ( x, y )=( R α j ( x ) ,yS α j ( x )) .