Torsion Points - Elliptic Curves: Number Theory and Cryptography - page 92

nR 2 = Q 2 .Then n ( R 1 + R 2 )= Q 1 + Q 2 ,so

τ n ( P, Q 1 + Q 2 )= e n ( P, R 1 + R 2 − φR 1 − φR 2 )
= e n ( P, R 1 − φR 1 ) e n ( P, R 2 − φR 2 )
= τ n ( P, Q 1 ) τ n ( P, Q 2 ) .

It remains to prove the nondegeneracy. This we postpone to Section 11.7.

The Tate-Lichtenbaum pairing can be used in some situations where the
Weil pairing does not apply. The Weil pairing needs E [ n ] ⊆ E ( F q ), which
implies that μ n ⊆ F q , by Corollary 3.11. The Tate-Lichtenbaum pairing

F q , but only needs a point of order n , rather than all
of E [ n ], to be in E ( F q ). In fact, it doesn't even need a point of order n .If
E ( F q )[ n ] is trivial, for example, then we have a pairing between two trivial
groups.

requires that μ n ⊆

Exercises
3.1 Let E be the elliptic curve y 2 = x 3 +1mod 5.

(a) Compute the division polynomial ψ 3 ( x ).
(b) Show that gcd( x 5 x, ψ 3 ( x )) = x .

(c) Use the result of part (b) to show that the 3-torsion points in E ( F 5 )
are {∞, (0 , 1) , (0 , − 1) } .

−

3.2 Let E be an elliptic curve in characteristic 2. Show that E [3] Z 3 ⊕ Z 3 .
( Hint: Use the formulas at the end of Section 2.8.)

3.3 Let E be an elliptic curve over a field of characteristic not 2. Let E [2] =
{∞ ,P 1 ,P 2 ,P 3 } . Show that e 2 ( P i ,P j )= − 1 whenever i = j .

3.4 Let M and N be 2 × 2 matrices with N = wx

.

N =

Define

yz

z −x
−yw

(this is the adjoint matrix).

(a) Show that Trace( M N )=det( M + N )

− det( M ) − det( N ).

(b) Use (a) to show that

b 2 det N
= ab (det( M + N ) − det M − det N )

a 2 det M det( aM + bN ) − −