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

Source 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 {k_0} (with char ( k 0 ) {\operatorname {char}}({k_0}) prime to m ), or over a local field k with residue field k 0 {k_0} , can be reduced to the computation of the discrete logarithm in k 0 ( ζ m ) ∗ {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 {k_0} which are divisible by m is reduced to the computation of the discrete logarithm in k 0 ( ζ m ) ∗ {k_0}{({\zeta _m})^ \ast } .