The triviality and nontriviality of Tate-Lichtenbaum self pairings on Jacobians of curves (Q2869044)

Pairings over abelian varieties have been widely used in cryptography, specially the Weil pairing \(e\) and the Tate-Lichtenbaum pairing \(t\) defined on a elliptic curve \(E\) over the finite field \(K:=\mathbb F_q\) of order \(q\). In this paper a fundamental difference between these pairings is studied: While \(e\) only admits trivial self pairings, \(t\) might do not; see for instance [\textit{I. F. Blake} (ed.) et al., Advances in elliptic curve cryptography. Lond. Math. Soc. Lect. Note Ser. 317. Cambridge: Cambridge University Press (2005; Zbl 1089.94018)]NEWLINENEWLINEAmong other interesting results, the author shows here that \(t\) only allows trivial self pairings if and only if the Frobenius morphism \(\Phi\) over \(K\) acts on \(E\) like an integer on \(E[n^2]\), the set of \(n^2\)-torsion points of \(E\), whenever \(E[n]\subseteq E(K)\) and \(\gcd(n,q)=1\). The condition on \(\Phi\) might be not easy to check in practice; the author then restate it by using complex multiplication. Finally, the author generalizes the results obtained in her thesis [PhD Thesis, University of Maryland, College Park, Maryland (2007)] to Jacobian of curves of arbitrary genus over finite fields.