Paper 2022/345

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

Wouter Castryck, KU Leuven, Ghent University
Marc Houben, KU Leuven, Leiden University
Frederik Vercauteren, KU Leuven
Benjamin Wesolowski, Institut de Mathématiques de Bordeaux, French Institute for Research in Computer Science and Automation

We show how the Weil pairing can be used to evaluate the assigned characters of an imaginary quadratic order $\mathcal{O}$ in an unknown ideal class $[\mathfrak{a}] \in \mathrm{Cl}(\mathcal{O})$ that connects two given $\mathcal{O}$-oriented elliptic curves $(E, \iota)$ and $(E', \iota') = [\mathfrak{a}](E, \iota)$. When specialized to ordinary elliptic curves over finite fields, our method is conceptually simpler and often faster than a recent approach due to Castryck, Sot\'akov\'a 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.

Available format(s)
Public-key cryptography
Publication info
Published elsewhere. ANTS XV -- Research in Number Theory
decisional Diffie-Hellman isogeny-based cryptography oriented elliptic curves class group action Weil pairing
Contact author(s)
wouter castryck @ esat kuleuven be
houben mr @ gmail com
frederik vercauteren @ esat kuleuven be
benjamin wesolowski @ math u-bordeaux fr
2022-10-01: last of 2 revisions
2022-03-14: received
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      author = {Wouter Castryck and Marc Houben and Frederik Vercauteren and Benjamin Wesolowski},
      title = {On the decisional Diffie-Hellman problem for class group actions on oriented elliptic curves},
      howpublished = {Cryptology ePrint Archive, Paper 2022/345},
      year = {2022},
      note = {\url{}},
      url = {}
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