Consider the jacobian of a hyperelliptic genus two curve defined over a
finite field. Under certain restrictions on the endomorphism ring of the
jacobian we give an explicit description all non-degenerate, bilinear,
anti-symmetric and Galois-invariant pairings on the jacobian. From this
description it follows that no such pairing can be computed more efficiently
than the Weil pairing.
To establish this result, we need an explicit description of the
representation of the Frobenius endomorphism on the l-torsion subgroup of the
jacobian. This description is given. In particular, we show that if the
characteristic polynomial of the Frobenius endomorphism splits into linear
factors modulo l, then the Frobenius is diagonalizable.
Finally, under the restriction that the Frobenius element is an element of a
certain subring of the endomorphism ring, we prove that if the characteristic
polynomial of the Frobenius endomorphism splits into linear factors modulo l,
then the embedding degree and the total embedding degree of the jacobian with
respect to l are the same number.Comment: Spelling errors correcte
