On the decisional Diffie–Hellman problem for class group actions on oriented elliptic curves
Wouter Castryck, Marc Houben, Fréderik Vercauteren, Benjamin Wesolowski
Abstract
Abstract We show how the Weil pairing can be used to evaluate the assigned characters of an imaginary quadratic order $${\mathcal {O}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>O</mml:mi></mml:math> in an unknown ideal class $$[{\mathfrak {a}}] \in {{\,\textrm{cl}\,}}({\mathcal {O}})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>[</mml:mo><mml:mi>a</mml:mi><mml:mo>]</mml:mo><mml:mo>∈</mml:mo><mml:mrow><mml:mspace/><mml:mtext>cl</mml:mtext><mml:mspace/></mml:mrow><mml:mo>(</mml:mo><mml:mi>O</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> that connects two given $${\mathcal {O}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>O</mml:mi></mml:math> -oriented elliptic curves $$(E, \iota )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo><mml:mi>E</mml:mi><mml:mo>,</mml:mo><mml:mi>ι</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> and $$(E', \iota ') = [{\mathfrak {a}}](E, \iota )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:msup><mml:mi>E</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>,</mml:mo><mml:msup><mml:mi>ι</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>[</mml:mo><mml:mi>a</mml:mi><mml:mo>]</mml:mo></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mi>E</mml:mi><mml:mo>,</mml:mo><mml:mi>ι</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> . 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.