**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.

**Category / Keywords: **implementation / Pairing-based cryptography

**Date: **received 5 Sep 2021, last revised 13 Sep 2021

**Contact author: **michael scott at tii ae

**Available format(s): **PDF | BibTeX Citation

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

**Version: **20210913:100239 (All versions of this report)

**Short URL: **ia.cr/2021/1130

[ Cryptology ePrint archive ]