Hyperelliptic Curves - Elliptic Curves: Number Theory and Cryptography - page 425

Take ( x − 1 , 0) and ( x +1 , − 1) as the factor base. Calculations yield

3 D 1 +5 D 2 =(( x +1) 2 , − x +1)=2( x +1 , − 1)

4 D 1 +3 D 2 =( x − 1 , 0)

D 1 +4 D 2 =( x 2 − 1 , − x +1)=( x − 1 , 0) + ( x +1 , − 1) .

If we take (first row) + 2(second row) − 2(third row), we obtain

9 D 1 +3 D 2 =0 .

Since the group J ( F 3 ) has order 10 (see Example 13.4), multiplication by 3
yields

7 D 1 = D 2 .

Exercises

13.1 Let C be the curve in Example 13.4. Use Cantor's algorithm to show
that ( x, i )+( x, − i )=(1 , 0).

13.2 Let E be the elliptic curve y 2 = x 3

− 2.

(a) Use Cantor's algorithm to compute the sum of pairs ( x − 3 , 5) +
( x − 3 , 5).
(b) Compute the sum (3 , 5) + (3 , 5) on E . Compare with (a). More
generally, see Exercise 13.9 below.

13.3 Let C be the hyperelliptic curve y 2 = x 5 − 5 x 3 +4 x +1.

(a) Show that div( y − 1) = [( − 1 , 1)] + [( − 2 , 1)] + [(1 , 1)] + [(2 , 1)] +
[(0 , 1)] − 5[ ∞ ]
(b) Show that div( x )=[(0 , 1)] + [(0 , ].

(c) Find a reduced divisor equivalent modulo principal divisors to
[( − 1 , 1)] + [( − 1 , − 1)] + [(1 , 1)] + [(2 , 1)] + [(0 , 1)] − 5[ ∞ ].

− 1)] − 2[ ∞

13.4 (a) Let F ( x, y ) be a function on a hyperelliptic curve and let G ( x, y )=

F ( x, −y ). What is the relation between div( F ) and div( G )?
(b) Let D be a principal divisor. Show that w ( D ) is also a principal
divisor. Give two proofs, one using (a) and the second using the
fact that D + w ( D ) is principal.

13.5 Let ( U, V ) be the pair corresponding to a semi-reduced divisor D . Show
that ( U, −V ) is the pair for w ( D ).