Torsion Points - Elliptic Curves: Number Theory and Cryptography - page 82

applications. However, to get rid of the extra 2 in the denominator, we proceed
as follows. Induction (treating separately the various possibilities for n mod
4) shows that

( x 2 + A ) ( n 2

− 1) / 4

ψ n ≡ (mod 2) when n is odd

and

n

2 ( x 2 + A ) ( n 2

(2 y ) − 1 ψ n ≡

− 4) / 4

(mod 2) when n is even .

A straightforward calculation now yields the lemma.

We now consider an elliptic curve

y 2 = x 3 + Ax + B, 4 A 3 +27 B 2

E : =0 .

We don't specify what ring or field the coe cients A, B are in, so we continue
to treat them as variables. We regard the polynomials in Z [ x, y 2 ,A,B ]as
polynomials in Z [ x, A, B ]byreplacing y 2 with x 3 + Ax + B . Therefore, we
write φ n ( x )and ψ n ( x ). Note that ψ n is not necessarily a polynomial in x
alone, while ψ n is always a polynomial in x .

LEMMA 3.5

φ n ( x )= x n 2 + low er degree term s
ψ n ( x )= n 2 x n 2

− 1 + low er degree term s

PROOF In fact, we claim that

ψ n = y ( nx ( n 2

− 4) / 2 +

··· )if n is even

nx ( n 2

− 1) / 2 +

··· if n is odd.

This is proved by induction. For example, if n =2 m +1 with m even, then
the leading term of ψ m +2 ψ 3 m is

( m +2) m 3 y 4 x ( m +2) 2 − 4

+ 3 m 2

− 12

.

2 2

Changing y 4 to ( x 3 + Ax + B ) 2 yields

( m +2) m 3 x (2 m +1) 2 − 1 .

2

Similarly, the leading term of ψ m− 1 ψ 3 m +1 is

( m − 1)( m +1) 3 x (2 m +1) 2 − 1 .

2