A note on group membership tests for $\G_1$, $\G_2$ and $\G_T$ on BLS pairing-friendly curves

Michael Scott

Paper 2021/1130

A note on group membership tests for $\G_{1}$ , $\G_{2}$ and $\G_{T}$ on BLS pairing-friendly curves

Michael Scott

Abstract

Here we consider a method for quickly testing for group membership in the groups $\G_{1}$ , $\G_{2}$ and $\G_{T}$ (all of prime order $r$ ) as they arise on a type-3 pairing-friendly curve. As is well known endomorphisms exist for each of these groups which allows for faster point multiplication for elements of order $r$ . The endomorphism applies if an element is of order $r$ . Here we show that, under relatively mild conditions, the endomorphism applies {\bf if and only if} an element is of order $r$ . This results in a faster method of confirming group membership. In particular we show that the conditions are met for the popular BLS family of curves.

Note: Fixed typo. Added observation on G2

Metadata

Available format(s): PDF
Category: Implementation
Publication info: Preprint.
Keywords: Pairing-based cryptography
Contact author(s): michael scott @ tii ae
History: 2022-10-01: last of 3 revisions; 2021-09-06: received; See all versions
Short URL: https://ia.cr/2021/1130
License: CC BY

BibTeX

@misc{cryptoeprint:2021/1130,
      author = {Michael Scott},
      title = {A note on group membership tests for $\G_1$, $\G_2$ and $\G_T$ on {BLS} pairing-friendly curves},
      howpublished = {Cryptology {ePrint} Archive, Paper 2021/1130},
      year = {2021},
      url = {https://eprint.iacr.org/2021/1130}
}

Paper 2021/1130

A note on group membership tests for \G1, \G2 and \GT on BLS pairing-friendly curves

Abstract

Metadata

A note on group membership tests for $\G_{1}$ , $\G_{2}$ and $\G_{T}$ on BLS pairing-friendly curves