Paper 2020/1070
Efficient indifferentiable hashing to elliptic curves $y^2 = x^3 + b$ provided that $b$ is a quadratic residue
Dmitrii Koshelev
Abstract
Let $\mathbb{F}_{\!q}$ be a finite field and $E_b\!: y^2 = x^3 + b$ be an ordinary elliptic $\mathbb{F}_{\!q}$-curve of $j$-invariant $0$ such that $\sqrt{b} \in \mathbb{F}_{\!q}$. In particular, this condition is fulfilled for the curve BLS12-381 and for one of sextic twists of the curve BW6-761 (in both cases $b=4$). These curves are very popular in pairing-based cryptography. The article provides an efficient constant-time encoding $h\!: \mathbb{F}_{\!q} \to E_b(\mathbb{F}_{\!q})$ of an absolutely new type for which $q/6 \leqslant \#\mathrm{Im}(h)$. We prove that at least for $q \equiv 4 \ (\mathrm{mod} \ 9)$ the hash function $H\!: \{0,1\}^* \to E_b(\mathbb{F}_{\!q})$ induced by $h$ is indifferentiable from a random oracle. The main idea of our encoding consists in extracting in $\mathbb{F}_{\!q}$ (for $q \equiv 1 \ (\mathrm{mod} \ 3)$) a cubic root instead of a square root as in the well known (universal) SWU encoding and in its simplified analogue. Besides, the new hashing can be implemented without quadratic and cubic residuosity tests (as well as without inversions) in $\mathbb{F}_{\!q}$. Thus in addition to the protection against timing attacks, $H$ is much more efficient than the SWU hash function, which generally requires to perform $4$ quadratic residuosity tests in $\mathbb{F}_{\!q}$. For instance, in the case of BW6-761 this allows to avoid approximately $4 \!\cdot\! 761 \approx 3000$ field multiplications.
Metadata
- Available format(s)
-
PDF
- Category
- Implementation
- Publication info
- Preprint.
- Keywords
- cubic residue symbol and cubic rootselliptic surfaceshashing to elliptic curvesindifferentiability from a random oraclepairing-based cryptography
- Contact author(s)
- dishport @ ya ru
- History
- 2021-06-18: last of 7 revisions
- 2020-09-09: received
- See all versions
- Short URL
- https://ia.cr/2020/1070
- License
-
CC BY
BibTeX
@misc{cryptoeprint:2020/1070, author = {Dmitrii Koshelev}, title = {Efficient indifferentiable hashing to elliptic curves $y^2 = x^3 + b$ provided that $b$ is a quadratic residue}, howpublished = {Cryptology {ePrint} Archive, Paper 2020/1070}, year = {2020}, url = {https://eprint.iacr.org/2020/1070} }