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

Authors: Gerhard Frey and Hans-Georg Rück
Journal: Math. Comp. 62 (1994), 865-874
MSC: Primary 11G20; Secondary 14G15, 94A60
MathSciNet review: 1218343
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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 }$.

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Article copyright: © Copyright 1994 American Mathematical Society