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Journal Articles LMS Journal of Computation and Mathematics Year : 2015

Heuristics on pairing-friendly abelian varieties

John L Boxall
  • Function : Author
  • PersonId : 961487
David Gruenewald
  • Function : Author

Abstract

We discuss heuristic asymptotic formulae for the number of isogeny classes of pairing-friendly abelian varieties of fixed dimension g 2 over prime finite fields. In each formula, the embedding degree k 2 is fixed and the rho-value is bounded above by a fixed real ρ0 > 1. The first formula involves families of ordinary abelian varieties whose endomorphism ring contains an order in a fixed CM-field K of degree g and generalizes previous work of the first author when g = 1. It suggests that, when ρ0 < g, there are only finitely many such isogeny classes. On the other hand, there should be infinitely many such isogeny classes when ρ0 > g. The second formula involves families whose endomorphism ring contains an order in a fixed totally real field K + 0 of degree g. It suggests that, when ρ0 > 2g/(g + 2) (and in particular when ρ0 > 1 if g = 2), there are infinitely many isogeny classes of g-dimensional abelian varieties over prime fields whose endomorphism ring contains an order of K + 0. We also discuss the impact that polynomial families of pairing-friendly abelian varieties has on our heuristics, and review the known cases where they are expected to provide more isogeny classes than predicted by our heuristic formulae.

Dates and versions

hal-02153416 , version 1 (14-06-2019)

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John L Boxall, David Gruenewald. Heuristics on pairing-friendly abelian varieties. LMS Journal of Computation and Mathematics, 2015, 18 (1), pp.419-443. ⟨10.1112/S1461157015000091⟩. ⟨hal-02153416⟩
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