Cryptography Reference
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have nonzero derivative. In characteristic p> 0, the polynomials with zero
derivative are exactly those of the form g ( x p ).)
Example 2.5
We continue with the previous example, where α ( P )=2 P .Wehave
R 1 ( x, y )= 3 x 2 + A
2
2 x.
2 y
The fact that y 2 = x 3 + Ax + B , plus a little algebraic manipulation, yields
r 1 ( x )= x 4
2 Ax 2
8 Bx + A 2
4( x 3 + Ax + B )
.
(This is the same as the expression in terms of division polynomials that will
be given in Section 3.2.) Therefore, deg( α ) = 4. The polynomial q ( x )=
4(3 x 2 + A ) is not zero (including in characteristic 3, since if A =0then
x 3 + B has multiple roots, contrary to assumption). Therefore α is separable.
Example 2.6
Let's repeat the previous example, but in characteristic 2. We'll use the
formulas from Section 2.8 for doubling a point. First, let's look at y 2 + xy =
x 3 + a 2 x 2 + a 6 .Wehave
α ( x, y )=( r 1 ( x ) ,R 2 ( x, y ))
with r 1 ( x )=( x 4 + a 6 ) /x 2 . Therefore deg( α ) = 4. Since p ( x )=4 x 3 =0and
q ( x )=2 x = 0, the endomorphism α is not separable.
Similarly, in the case y 2 + a 3 y = x 3 + a 4 x + a 6 ,wehave r 1 ( x )=( x 4 + a 4 ) /a 3 .
Therefore, deg( α ) = 4, but α is not separable.
In general, in characteristic p ,themap α ( Q )= pQ has degree p 2 and is not
separable. The statement about the degree is Corollary 3.7. The fact that α
is not separable is proved in Proposition 2.28.
An important example of an endomorphism is the Frobenius map . Sup-
pose E is defined over the finite field F q .Let
φ q ( x, y )=( x q ,y q ) .
The Frobenius map φ q plays a crucial role in the theory of elliptic curves over
F q .
LEMMA 2.20
Let E be defined over F q .Then φ q is an endom orphism
of E of degree q ,
and φ q is not separable.
 
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