### 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

Available format(s)
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
See all versions
Short URL
https://ia.cr/2021/1130

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}
}

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