Paper 2026/001

The Cokernel Pairing

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.