Families of Prime-Order Endomorphism-Equipped Embedded Curves on Pairing-Friendly Curves

This paper presents a procedure to construct parameterized families of prime-order endomorphism-equipped elliptic curves that are defined over the scalar field of pairing-friendly elliptic curve families such as Barreto-Lynn-Scott (BLS), Barreto-Naehrig (BN) and Kachisa-Schaefer-Scott (KSS), providing general formulas derived from the curves' seeds. These so-called "embedded curves" are of major interest in SNARK applications that prove statements involving elliptic curve arithmetic i.e. digital signatures. In this paper, the mathematical groundwork is laid, and advantages of these embeddings are discussed. Additionally, practical examples in the case of BN and BLS families are included and impossibility results regarding KSS families are explained.