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: Some further simplification. Proper latex rendering of point-at infinity

Metadata
Available format(s)
PDF
Category
Implementation
Publication info
Preprint. MINOR revision.
Keywords
Pairing-based cryptography
Contact author(s)
michael scott @ tii ae
History
2021-09-13: last of 2 revisions
2021-09-06: received
See all versions
Short URL
https://ia.cr/2021/1130
License
Creative Commons Attribution
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},
      note = {\url{https://eprint.iacr.org/2021/1130}},
      url = {https://eprint.iacr.org/2021/1130}
}
Note: In order to protect the privacy of readers, eprint.iacr.org does not use cookies or embedded third party content.