Divisors - Elliptic Curves: Number Theory and Cryptography - page 373

By Weil reciprocity (Lemma 11.11), h 2 (div( h 1 )) = h 1 (div( h 2 )). Also, Weil
reciprocity yields

h 2 ( D S ) n = h 2 ( nD S )= h 2 (div( F S )) = F S (div( h 2 ))

and similarly h 1 ( D T ) n = F T (div( h 1 )). Therefore, we obtain

F S ( D T ) = F T ( D S )

F T ( D S )

F S ( D T ) = e n ( S, T ) .

If D S and D S are not necessarily disjoint from D T and D T , we can proceed

in two steps. First, let

D S =[ X 1 + S ] − [ X 1 ] , D T

=[ Y 1 + T ] − [ Y 1 ] ,

where X 1 and Y 1 are chosen so that D S and D S are disjoint from D T and

D T and so that D S and D S are disjoint from D T and D T . The preceding

argument shows that

F S ( D T ) = F T ( D S )

F T ( D S ) F T ( D S )

F S ( D T ) = F S ( D T ) = e n ( S, T ) .

This completes the proof.

For other proofs, see [69, Section 6.4] and [51].

11.6.2 The Tate-Lichtenbaum Pairing

In Section 3.4, we defined the (modified) Tate-Lichtenbaum pairing in terms
of the Weil pairing. In Section 11.3, we gave an alternative definition in terms
of divisors.

THEOREM 11.25
Thepairings τ n defined in T heorem 3.17 and T heorem 11.8areequa .

PROOF Let the notation be as in Theorem 3.17. In particular, Q ∈ E ( F q )
and nR = Q . Choose a function g such that

div( g )= n [ R ] − [ Q ] − ( n − 1)[ ∞ ] .

Let g φ denote the function obtained by applying φ to all the coe cients o f t he
rational function defining g ,so φ ( g ( X )) = g φ ( φX ) for all points X ∈ E ( F q ).
Since φ ( Q )= Q ,

div( g φ )= n [ φ ( R )] − [ Q ] − ( n − 1)[ ∞ ] .