**Hashing to elliptic curves of $j=0$ and quadratic imaginary orders of class number $2$**

*Dmitrii Koshelev*

**Abstract: **Let $\mathbb{F}_{\!p}$ be a prime finite field ($p > 5$) and $E_b\!: y_0^2 = x_0^3 + b$ be an elliptic $\mathbb{F}_{\!p}$-curve of $j$-invariant $0$. In this article we produce the simplified SWU hashing to curves $E_b$ having an $\mathbb{F}_{\!p^2}$-isogeny of degree $5$. This condition is fulfilled for some Barreto--Naehrig curves, including BN512 from the standard ISO/IEC 15946-5. Moreover, we show how to implement the simplified SWU hashing in constant time (for any $j$-invariant), namely without quadratic residuosity tests and inversions in $\mathbb{F}_{\!p}$. Thus in addition to the protection against timing attacks, the new hashing $h\!: \mathbb{F}_{\!p} \to E_b(\mathbb{F}_{\!p})$ turns out to be much more efficient than the (universal) SWU hashing, which generally requires to perform $2$ quadratic residuosity tests.

**Category / Keywords: **implementation / constant-time implementation, hashing to elliptic curves, Kummer surfaces, pairing-based cryptography, rational curves and their parametrization, vertical isogenies

**Date: **received 8 Aug 2020, last revised 4 Dec 2020

**Contact author: **dishport at yandex ru

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

**Version: **20201204:111634 (All versions of this report)

**Short URL: **ia.cr/2020/969

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