Paper 2026/137 Original: eprint.iacr.org/2026/137 ↗

Hensel-lifting black-box algorithms and fast trace computation for elliptic-curve endomorphisms

Abstract

We demonstrate a general and efficient technique to Hensel-lift a solution to a system of ($p$‑adically analytic) equations which may be given implicitly in the form of an efficient evaluation algorithm. Contrary to textbook Hensel lifting, we do not require the equations to be represented explicitly; indeed, our main application uses the method for a system of equations that can be exponentially larger than its representation as an arithmetic circuit: we show how to compute traces of separable elliptic-curve endomorphisms over a finite field $\mathbb{F}_q$ by constructing an approximate lift to $\mathbb{Z}_q$. Our examples include endomorphisms represented as a chain of Vélu, √élu, modular, or radical isogenies, as well as HD‑embedded endomorphisms. The resulting trace-computation algorithm outperforms the state of the art both asymptotically and concretely.