Paper 2022/037

Subgroup membership testing on elliptic curves via the Tate pairing

Dmitrii Koshelev

Abstract

This note explains how to guarantee the membership of a point in the prime order subgroup of an elliptic curve (over a finite field) satisfying some moderate conditions. For this purpose, we apply the Tate pairing on the curve, however it is not required to be pairing-friendly. Whenever the cofactor is small, the given approach is more efficient than other known ones, because it needs to compute at most two $n$-th power residue symbols (with small $n$) in the basic field. In particular, we deal with two Legendre symbols for the curve Bandersnatch proposed by the Ethereum Foundation team. Due to recent improvements of Euclidean type constant-time algorithms for the Legendre symbol computation, the new subgroup check is almost free for that curve.