A remark concerning $m$-divisibility and the discrete logarithm in the divisor class group of curves

Abstract:

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 ${k_0}$ (with ${\operatorname {char}}({k_0})$ prime to m), or over a local field k with residue field ${k_0}$, can be reduced to the computation of the discrete logarithm in ${k_0}{({\zeta _m})^ \ast }$. 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 ${k_0}$ which are divisible by m is reduced to the computation of the discrete logarithm in ${k_0}{({\zeta _m})^ \ast }$.