**Finite field mapping to elliptic curves of $j$-invariant $1728$**

*Dmitrii Koshelev*

**Abstract: **This article generalizes the simplified Shallue--van de Woestijne--Ulas (SWU) method of deterministic finite field mapping $\mathbb{F}_{\!q} \to E(\mathbb{F}_{\!q})$ to the case of any elliptic $\mathbb{F}_{\!q}$-curve $E$ of $j$-invariant $1728$. More precisely, we obtain a rational $\mathbb{F}_{\!q}$-curve $C$ (and its explicit quite simple proper $\mathbb{F}_{\!q}$-parametrization $par\!: \mathbb{P}^1 \to C$) on the Kummer surface $K$ associated with the direct product $E \!\times\! E^\prime$, where $E^\prime$ is the quadratic $\mathbb{F}_{\!q}$-twist of $E$. The SWU method consists in computing the direct image of $par$ and a subsequent inverse image $(P,Q)$ of the natural two-sheeted covering $\rho\!: E \!\times\! E^\prime \to K$. Denoting by $\sigma\!:E^\prime \to E$ the corresponding $\mathbb{F}_{\!q^2}$-isomorphism, it is easily seen that $P \in E(\mathbb{F}_{\!q})$ or $\sigma(Q) \in E(\mathbb{F}_{\!q})$. We produce the curve $C$ as one of two absolutely irreducible $\mathbb{F}_{\!q}$-components of $pr^{{-}1}(C_8)$ for some rational $\mathbb{F}_{\!q}$-curve $C_8$ of bidegree $(8,8)$ with $42$ singular points, where $pr\!: K \to \mathbb{P}^1 \!\times\! \mathbb{P}^1$ is the two-sheeted projection to $x$-coordinates of $E$ and $E^\prime$.

**Category / Keywords: **implementation / finite fields, pairing-based cryptography, elliptic curves of $j$-invariant $1728$, Kummer surfaces, rational curves, Weil restriction, isogenies

**Date: **received 7 Nov 2019, last revised 21 Dec 2019

**Contact author: **dishport at ya ru

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

**Version: **20191221:092317 (All versions of this report)

**Short URL: **ia.cr/2019/1294

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