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 {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 } .