The Basic Theory - Elliptic Curves: Number Theory and Cryptography - page 58

PROPOSITION 2.28
Let E be an elliptic curve defined over a field K ,and et n be a n on zero
integer. Suppose that m ultiplication by n on E isgiven by

n ( x, y )=( R n ( x ) ,yS n ( x ))

for all ( x, y )

∈ E ( K ) ,where R n and S n are rationalfunctions. T hen

R n ( x )
S n ( x )

= n.

T herefore, m ultiplication by n is separableifand onlyif n isnotamu tiple
of the characteristic p of the field.

S n ,wehave R −n /S −n = R n /S n . PROOF Since R −n = R n and S −n = − −

Therefore, the result for positive n implies the result for negative n .
Note that the first part of the proposition is trivially true for n =1. Ifit
is true for n ,thenLemma2.26impliesthatitistruefor n +1, which is the
sum of n and 1. Therefore,

R n ( x )

S ( x ) = n for all n .
We have R n ( x ) =0ifandonlyif n = R n ( x ) /S n ( x ) = 0, which is equivalent
to p not dividing n . Since the definition of separability is that R n =0,this

proves the second part of the proposition.

Finally, we use Lemma 2.26 to prove a result that will be needed in Sec-
tions 3.2 and 4.2. Let E be an elliptic curve defined over a finite field F q .
The Frobenius endomorphism φ q is defined by φ q ( x, y )=( x q ,y q ). It is an
endomorphism of E by Lemma 2.20.

PROPOSITION 2.29
Let E be an elliptic curve defined over F q ,where q isapowerofthe prime p .
Let r and s be integers, not both 0 . T he endom orphism rφ q + s is separableif
and onlyif p s .

PROOF Write the multiplication by r endomorphism as

r ( x, y )=( R r ( x ) ,yS r ( x )) .

Then

( R rφ q ( x ) ,yS rφ q ( x )) = ( φ q r )( x, y )=( R r ( x ) ,y q S r ( x ))
= R r ( x ) ,y ( x 3 + Ax + B ) ( q− 1) / 2 S r ( x ) .

Therefore,

c rφ q = R rφ q /S rφ q = qR q− 1 R r /S rφ q =0 .

r