Divisors - Elliptic Curves: Number Theory and Cryptography - page 357

Therefore, the pairing is independent mod n th powers of the choice of D P
and D Q .
If Q 1 and Q 2 are two points and D Q 1 and D Q 2 are corresponding divisors,
then

D Q 1 + D Q 2 ∼ [ Q 1 ] − [ ∞ ]+[ Q 2 ] − [ ∞ ] ∼ [ Q 1 + Q 2 ] − [ ∞ ]

where ∼ denotes equivalence of divisors mod principal divisors. The last
equivalence is the fact that the sum function in Corollary 11.4 is a homomor-
phism of groups. Consequently,

P, Q 1 + Q 2 n = f ( D Q 1 ) f ( D Q 2 )= P, Q 1 n P, Q 2 n .

Therefore, the pairing is linear in the second variable.
If P 1 ,P 2 ∈ E ( F q )[ n ], and D P 1 ,D P 2 are corresponding divisors and f 1 ,f 2

are the corresponding functions, then

D P 1 + D P 2 ∼ [ P 1 ] − [ ∞ ]+[ P 2 ] − [ ∞ ] ∼ [ P 1 + P 2 ] − [ ∞ ] .

Therefore, we can let D P 1 + P 2 = D P 1 + D P 2 .Wehave

div( f 1 f 2 )= nD P 1 + nD P 2 = nD P 1 + P 2 ,

so f 1 f 2 can be used to compute the pairing. Therefore,

P 1 + P 2 ,Q n = f 1 ( D Q ) f 2 ( D Q )= P 1 ,Q n P 2 ,Q n .

Consequently, the pairing is linear in the first variable.
The nondegeneracy is much more di cult to prove. This will follow from
the main results of Sections 11.7 and 11.6.2; namely, the present pairing is the
same as the pairing defined in Chapter 3, and that pairing is nondegenerate.
Since F q is cyclic of order q − 1, the ( q − 1) /n -th power map gives an

isomorphism

F q / ( F q ) n −→ μ n .

Therefore, define

( q − 1) /n

τ n ( P, Q )= P, Q .

n

The desired properties of the modified Tate-Lichtenbaum pairing τ n follow
immediately from those of the Tate-Lichtenbaum pairing.

11.4 Computation of the Pairings

In Section 11.1, we showed how to express a divisor of degree 0 and sum
∞ as a divisor of a function. This method suces to compute the Weil and
Tate-Lichtenbaum pairings for small examples. However, for larger examples,