On the decisional Diffie–Hellman problem for class group actions on oriented elliptic curves

Research in Number Theory, vol. 8

Abstract

Abstract We show how the Weil pairing can be used to evaluate the assigned characters of an imaginary quadratic order $${\mathcal {O}}$$ O in an unknown ideal class $$[{\mathfrak {a}}] \in {{\,\textrm{cl}\,}}({\mathcal {O}})$$ [a]∈cl(O) that connects two given $${\mathcal {O}}$$ O -oriented elliptic curves $$(E, \iota )$$ (E,ι) and $$(E', \iota ') = [{\mathfrak {a}}](E, \iota )$$ (E′,ι′)=[a](E,ι) . When specialized to ordinary elliptic curves over finite fields, our method is conceptually simpler and often somewhat faster than a recent approach due to Castryck, Sotáková and Vercauteren, who rely on the Tate pairing instead. The main implication of our work is that it breaks the decisional Diffie–Hellman problem for practically all oriented elliptic curves that are acted upon by an even-order class group. It can also be used to better handle the worst cases in Wesolowski’s recent reduction from the vectorization problem for oriented elliptic curves to the endomorphism ring problem, leading to a method that always works in sub-exponential time.

Authors 4

1. Wouter Castryck

   Ghent University · IMEC · KU Leuven

   Affiliation as printed

   Dept. Mathematics: Algebra and Geometry, Ghent University, Krijgslaan 281, 9000, Gent, Belgium

   imec-COSIC, KU Leuven, Kasteelpark Arenberg ,10/2452, 3001, Leuven, Belgium

2. Marc Houben corresponding Aachen Dept. Mathematics

   Leiden University · IMEC · KU Leuven

   Affiliation as printed

   Dept. Mathematics, KU Leuven, Celestijnenlaan 200B, 3001, Leuven, Belgium

   Dept. Mathematics, Leiden Univ, Niels Bohrweg 1, 2333 CA, Leiden, The Netherlands

   imec-COSIC, KU Leuven, Kasteelpark Arenberg ,10/2452, 3001, Leuven, Belgium

3. Fréderik Vercauteren

   IMEC · KU Leuven

   Affiliation as printed

   imec-COSIC, KU Leuven, Kasteelpark Arenberg ,10/2452, 3001, Leuven, Belgium

4. Benjamin Wesolowski

   Centre National de la Recherche Scientifique · Institut national de recherche en sciences et technologies du numérique · Université de Bordeaux · Institut Polytechnique de Bordeaux · Institut de Mathématiques de Bordeaux

   Affiliation as printed

   INRIA, IMB, UMR 5251, 33400, Talence, France

   Univ. Bordeaux, CNRS, Bordeaux INP, IMB, UMR 5251, 33400, Talence, France