Families of prime-order endomorphism-equipped embedded curves on pairing-friendly curves

Antonio Sanso; Youssef El Housni

Paper 2023/1662

Families of prime-order endomorphism-equipped embedded curves on pairing-friendly curves

Antonio Sanso

, Ethereum Foundation

Youssef El Housni

, Linea

Abstract

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.

Metadata

Available format(s): PDF
Category: Public-key cryptography
Publication info: Preprint.
Keywords: elliptic curves bilinear pairings complex multiplication zeroknowledge proofs
Contact author(s): asanso @ ethereum org
youssef elhousni @ consensys net
History: 2024-05-09: last of 6 revisions; 2023-10-26: received; See all versions
Short URL: https://ia.cr/2023/1662
License: CC BY

BibTeX

@misc{cryptoeprint:2023/1662,
      author = {Antonio Sanso and Youssef El Housni},
      title = {Families of prime-order endomorphism-equipped embedded curves on pairing-friendly curves},
      howpublished = {Cryptology {ePrint} Archive, Paper 2023/1662},
      year = {2023},
      url = {https://eprint.iacr.org/2023/1662}
}