1970-01-01

(2026-01) The Cokernel Pairing

Krijn Reijnders

2026-01-01

Abstract

We study a new pairing, beyond the Weil and Tate pairing. The Weil pairing is a non-degenerate pairing $E[m] \times E[m] \to \mu_{m}$ , which operates on the kernel of $[m]$ . Similarly, when $\mu_{m} \subseteq \mathbb{F}_q^*$ , the Tate pairing is a non-degenerate pairing $E[m](\mathbb{F}_q) \times E(\mathbb{F}_q) / [m]E(\mathbb{F}_q) \to \mu_{m}$ , which connects the kernel and the rational cokernel of $[m]$ . We define a pairing [ \langle{\quad}\rangle_m : E(\mathbb{F}_q) / [m]E(\mathbb{F}_q) \times E(\mathbb{F}_q) / [m]E(\mathbb{F}q) \to \mu{m}] on the rational cokernels of $[m]$ , filling the gap left by the Weil and Tate pairing. When $E[m] \subseteq E(\mathbb{F}_q)$ , this pairing is non-degenerate, and can be computed using three Tate pairings, and two discrete logarithms in $\mu_{m}$ , assuming a basis for $E[m]$ . For $m = \ell$ prime, this pairing allows us to study $E(\mathbb{F}_q) / [\ell]E(\mathbb{F}_q)$ directly and to simplify the computation for a basis of $E[\ell^k]$ , and more generally the Sylow $\ell$ -torsion. This finds natural applications in isogeny-based cryptography when computing $\ell^k$ -isogenies.