A REMARK CONCERNING M-DIVISIBILITY AND THE DISCRETE LOGARITHM IN THE DIVISOR CLASS GROUP OF CURVES

The aim of this paper is to show that the computation of the discrete logarithm in the m-torsion part of the divisor class group of a curve X over a finite field k0 (with char(k0) prime to m), or over a local field k with residue field k0, can be reduced to the computation of the discrete logarithm in k0(zetam)* . For this purpose we use a variant of the (tame) Tate pairing for Abelian varieties over local fields. In the same way the problem to determine all linear combinations of a finite set of elements in the divisor class group of a curve over k or k0 which are divisible by m is reduced to the computation of the discrete logarithm in k0(zetam)*.