Torsion Points - Elliptic Curves: Number Theory and Cryptography - page 93

for all scalars a, b . This is the relation used in the proof of Propo-
sition 3.16.

3.5 Show that part (6) of Theorem 3.9 holds when α is the endomorphism
given by multiplication by an integer m .

3.6 Let E be an elliptic curve over a field K and let P be a point of order
n (where n is not divisible by the characteristic of the field K ). Let
Q ∈ E [ n ]. Show that there exists an integer k such that Q = kP if and
only if e n ( P, Q )=1.

3.7 Write the equation of the elliptic curve E as

F ( x, y, z )= y 2 z − x 3 − Axz 2 − Bz 3 =0 .

Show that a point P on E is in E [3] if and only if

⎛ ⎞

F xx F xy F xz
F yx F yy F yz
F zx F zy F zz

⎝ ⎠ =0

det

at the point P ,where F ab denotes the 2nd partial derivative with respect
to a, b . The determinant is called the H essian .Foracurvein P 2 defined
by an equation F = 0, a point where the Hessian is zero is called a flex
of the curve.

3.8 The division polynomials ψ n were defined for n 0. Show that if we

let ψ −n = −ψ n , then the recurrence relations preceding Lemma 3.3,
which are stated only for m ≥ 2, hold for all integers m . (Note that this
requires verifying the relations for m ≤− 2 and for m = − 1 , 0 , 1.)

≥