## Cryptology ePrint Archive: Report 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.

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

Short URL: ia.cr/2021/1130

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