Complex Multiplication - Elliptic Curves: Number Theory and Cryptography - page 320

Example 10.3
Let E be the elliptic curve defined over F 2 by

y 2 + y = x 3 .

An easy calculation shows that E ( F 2 ) consists of 3 points, so

a =2+1 − # E ( F 2 )=2+1 − 3=0 .

Therefore, E is supersingular and the Frobenius endomorphism φ 2 satisfies
φ 2 +2=0 .

If ( x, y ) ∈ E ( F 2 ), then

2( x, y )= −φ 2 ( x, y )= − ( x 4 ,y 4 )=( x 4 ,y 4 +1) ,

since negation on E is given by

− ( x, y )=( x, y +1) .

By Theorem 10.6, the endomorphism ring is a maximal order in a quaternion
algebra ramified at only 2 and ∞ . We gave such a maximal order in (10.2)
above. Let's start by finding endomorphisms corresponding to i , j , k .Let
ω ∈ F 4 satisfy

ω 2 + ω +1=0 .
Define endomorphisms i , j , k by

i ( x, y )=( x +1 ,y + x + ω )
j ( x, y )=( x + ω, y + ω 2 x + ω )
k ( x, y )=( x + ω 2 ,y + ωx + ω ) .

An easy calculation shows that

i ( j ( x, y )) = k ( x, y ) , j ( i ( x, y )) = − k ( x, y )

and that

i 2 = k 2 = k 2 = − 1 .

A straightforward calculation yields

(1 + i + j + k )( x, y )=( ωx 4 ,y 4 )= φ 2 ( ωx, y )=

− 2( ω ( x, y )) ,

where ω is used to denote the endomorphism ( x, y ) → ( ωx, y ). Therefore,

1+ i + j + k
2

= −ω ∈ End( E ) .

It follows that

Z + Zi + Zj + Z 1+ i + j + k
2

⊆ End( E ) .

In fact, by Theorem 10.6, this is the whole endomorphism ring.