Paper 2025/196

Endomorphisms for Faster Cryptography on Elliptic Curves of Moderate CM Discriminants, II

Abstract

The present article is a natural extension of the previous one about the GLV method of accelerating a (multi-)scalar multiplication on elliptic curves of moderate CM discriminants $D<0$. In comparison with the first article, much greater magnitudes of $D$ (in absolute value) are achieved, although the base finite fields of the curves have to be pretty large. This becomes feasible by resorting to quite powerful algorithmic tools developed primarily in the context of lattice-based and isogeny-based cryptography. Curiously, pre-quantum cryptography borrows research outcomes obtained when seeking conversely quantum-resistant solutions or attacks on them. For instance, some $$-cycle of pairing-friendly MNT curves (with $$, i.e., $$) is relevant for the result of the current article. The given $$-cycle was generated at one time by Guillevic to provide $$ security bits, hence it was close to application in real-world zk-SNARKs. Another more performant MNT $$-cycle (with slightly smaller security level, but with much larger $$) was really employed in the protocol Coda (now Mina) until zero-knowledge proof systems on significantly faster pairing-free (or half-pairing) $$-cycles were invented. It is also shown in the given work that more lollipop curves, recently proposed by Costello and Korpal to replace MNT ones, are now covered by the GLV technique.