Embedded Elliptic Curves and Embedded Families for SNARK-Friendly Elliptic Curves

In 2021, Masson, Sanso, and Zhang introduced the Bandersnatch curve associated to the BLS12-381 pairing-friendly curve, an elliptic curve designed for zero-knowledge proofs requiring circuits with a curve arithmetic. This type of curve is useful for privacy-preserving protocols, and more generally for succinct validity proof using pairing-based SNARKs. An embedded curve is defined over a field whose order is the group order of its associated curve. In this way, the pairing-friendly curve is used to express a zero-knowledge proof (such as a SNARK) of a statement taking place on the embedded curve. Contrary to the previous embedded curves (such as CØCØ, JubJub), Bandersnatch was built with the complex multiplication (CM) method, in order to ensure a very small discriminant (-8, whose magnitude is small), and thus efficient scalar multiplication thanks to the GLV technique. The algorithm provided by Masson, Sanso, and Zhang for searching this type of curves requires computation of Hilbert class polynomials, making the search of curve slow. It was not known whether Bandersnatch was an exceptional curve or whether comparable curves exist, of larger discriminants. This paper highlights the technicalities of the CM method already in use in the 90s to generate curve parameters of chosen order. This old technique allows revisiting the curve search of Bandersnatch, providing a dramatic speed-up improvement. This paper presents two algorithms: one to generate embedded elliptic curves of SNARK-friendly elliptic curves, with a variable discriminant; a second to generate families (parameterized by polynomials) with a fixed discriminant. When the (negative) discriminant is -3 modulo 4, it is possible to obtain a prime-order curve, and form a cycle. To illustrate this, we apply the technique first to generate more embedded curves like Bandersnatch with BLS12-381, such as a curve of discriminant -6673027, defining a plain twist-secure cycle. We also comment on the scarcity of Bandersnatch-like CM curves, and recall that with this generic algorithm, it is only a question of core-hours to find them. Second, we show the link between a paper of Ben Smith in 2015 and the work of Dai, Lin, Zhao, and Zhou in 2023, obtaining prime-order parameterized families of embedded curves of fixed discriminant, such as -3 for BLS and KSS18 curves. With KSS16 curves, the discriminant -4 is also possible (the curve has an even order). The technique can work with any KSS, Scott–Guillevic, Gasnier–Guillevic, or other fixed-discriminant parameterized family of pairing-friendly curves. This paper provides a more general point of view on embedded curves such as Bandersnatch, putting into perspective the works of Masson, Sanso, and Zhang, and Sanso and El Housni. The Python/SageMath scripts are available at https://gitlab.inria.fr/zk-curves/cm-embedded-curves/.