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Media type:
E-Article
Title:
A remark concerning 𝑚-divisibility and the discrete logarithm in the divisor class group of curves
Contributor:
Frey, Gerhard;
Rück, Hans-Georg
imprint:
American Mathematical Society (AMS), 1994
Published in:Mathematics of Computation
Language:
English
DOI:
10.1090/s0025-5718-1994-1218343-6
ISSN:
0025-5718;
1088-6842
Origination:
Footnote:
Description:
<p>The aim of this paper is to show that the computation of the discrete logarithm in the <italic>m</italic>-torsion part of the divisor class group of a curve <italic>X</italic> over a finite field <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k 0">
<mml:semantics>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:msub>
<mml:mi>k</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
</mml:mrow>
<mml:annotation encoding="application/x-tex">{k_0}</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> (with <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c h a r left-parenthesis k 0 right-parenthesis">
<mml:semantics>
<mml:mrow>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mi>char</mml:mi>
</mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:msub>
<mml:mi>k</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
</mml:mrow>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
<mml:annotation encoding="application/x-tex">{\operatorname {char}}({k_0})</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> prime to <italic>m</italic>), or over a local field <italic>k</italic> with residue field <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k 0">
<mml:semantics>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:msub>
<mml:mi>k</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
</mml:mrow>
<mml:annotation encoding="application/x-tex">{k_0}</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>, can be reduced to the computation of the discrete logarithm in <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k 0 left-parenthesis zeta Subscript m Baseline right-parenthesis Superscript asterisk">
<mml:semantics>
<mml:mrow>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:msub>
<mml:mi>k</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
</mml:mrow>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:msub>
<mml:mi>ζ<!-- ζ --></mml:mi>
<mml:mi>m</mml:mi>
</mml:msub>
</mml:mrow>
<mml:msup>
<mml:mo stretchy="false">)</mml:mo>
<mml:mo>∗<!-- ∗ --></mml:mo>
</mml:msup>
</mml:mrow>
</mml:mrow>
<mml:annotation encoding="application/x-tex">{k_0}{({\zeta _m})^ \ast }</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>. 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 <italic>k</italic> or <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k 0">
<mml:semantics>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:msub>
<mml:mi>k</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
</mml:mrow>
<mml:annotation encoding="application/x-tex">{k_0}</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> which are divisible by <italic>m</italic> is reduced to the computation of the discrete logarithm in <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k 0 left-parenthesis zeta Subscript m Baseline right-parenthesis Superscript asterisk">
<mml:semantics>
<mml:mrow>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:msub>
<mml:mi>k</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
</mml:mrow>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:msub>
<mml:mi>ζ<!-- ζ --></mml:mi>
<mml:mi>m</mml:mi>
</mml:msub>
</mml:mrow>
<mml:msup>
<mml:mo stretchy="false">)</mml:mo>
<mml:mo>∗<!-- ∗ --></mml:mo>
</mml:msup>
</mml:mrow>
</mml:mrow>
<mml:annotation encoding="application/x-tex">{k_0}{({\zeta _m})^ \ast }</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>.</p>