Divisors - Elliptic Curves: Number Theory and Cryptography - page 377

LEMMA 11.30
Let A and B be finiteabe ian groups (w ritten additively) such that nx =0
for all x ∈ A and for all x ∈ B .Supposethat there is a nondegenerate(in
botharguments) bilinear pairing

, : B × A → μ n ,

where μ n isthe group of n throotsofunity(insomefie d). Let C be a su bgrou p
of B . D efine

ψ : A −→ μ n

c ∈ C

a −→ ( ··· , c, a , ··· ) .

Then

# ψ ( A )=# C.

PROOF The pairing is nondegenerate, so A Hom( B, μ n ). Clearly,

Ker ψ = { a ∈ A | c, a =1forall c ∈ C } .

Identifying A with the set of homomorphisms from B to μ n , we see that

Ker ψ = {f ∈ Hom( B, μ n ) | f ( C )=1 }.

But a homomorphism that sends C to1isexactlythesameasahomo-
morphism from B/C to μ n . The set of such homomorphisms has order
#( B/C )=# B/ # C . Therefore (see Theorem B.6 in Appendix B)

# ψ ( A )=# A/ #Ker ψ =# A/ #( B/C )=# C,

since # A =# B . This proves the lemma.

We can now apply the above to the elliptic curve E .Let

ψ : E [ n ] −→ μ n

P ∈ E ( F q )[ n ]

Q −→ ( ··· ,e n ( P, Q ) , ··· ) .

LEMMA 11.31
Let φ = φ q be the q th pow er Frobenius endom orphism of E .Then Ker ψ =

( φ − 1) E [ n ] .