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**2021/1082**

**Title:**

Some remarks on how to hash faster onto elliptic curves

**Authors:**
Dmitrii Koshelev

**Abstract:**

In this article we propose three optimizations of indifferentiable hashing onto (prime order subgroups of) ordinary elliptic curves over finite fields \mathbb{F}_{\!q}. One of them is dedicated to elliptic curves E provided that q \equiv 2 \ (\mathrm{mod} \ 3). The other two optimizations take place respectively for the subgroups \mathbb{G}_1, \mathbb{G}_2 of some pairing-friendly curves. The performance gain comes from the smaller number of required exponentiations in \mathbb{F}_{\!q} for hashing to E(\mathbb{F}_{\!q}), \mathbb{G}_2 (resp. from the absence of necessity to hash directly onto \mathbb{G}_1). In particular, our results affect the pairing-friendly curve BLS12-381 (the most popular in practice at the moment) as well as a few ones from the international draft NIST SP 800-186. Among other things, we present a taxonomy of state-of-the-art hash functions to elliptic curves.

**ePrint:**
https://eprint.iacr.org/2021/1082

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