Paper 2021/1446

Batch point compression in the context of advanced pairing-based protocols

Dmitrii Koshelev
Abstract

This paper continues author's previous ones about compression of points on elliptic curves $E_b\!: y^2 = x^3 + b$ (with $j$-invariant $0$) over a finite field $\mathbb{F}_{\!q}$. More precisely, we show in detail how any two (resp. three) points from $E_b(\mathbb{F}_{\!q})$ can be quickly compressed to two (resp. three) elements of $\mathbb{F}_{\!q}$ (apart from a few auxiliary bits) in such a way that the corresponding decompression stage requires to extract only one cubic (resp. sextic) root in $\mathbb{F}_{\!q}$ (with several multiplications and without inversions). As a result, for many $q$ occurring in practice the new compression-decompression methods are more efficient than the classical one with the two (resp. three) $x$ or $y$ coordinates of the points, which extracts two (resp. three) roots in $\mathbb{F}_{\!q}$. We explain why the new methods are useful in the context of modern real-world pairing-based protocols such as Groth16. As a by-product, when $q \equiv 2 \ (\mathrm{mod} \ 3)$ (in particular, $E_b$ is supersingular), we obtain a two-dimensional analogue of Boneh--Franklin's encoding, that is a way to sample two "independent'' $\mathbb{F}_{\!q}$-points on $E_b$ at the cost of one cubic root in $\mathbb{F}_{\!q}$. Finally, we comment on the case of four and more points from $E_b(\mathbb{F}_{\!q})$.

BibTeX

@misc{cryptoeprint:2021/1446,
      author = {Dmitrii Koshelev},
      title = {Batch point compression in the context of advanced pairing-based protocols},
      howpublished = {Cryptology ePrint Archive, Paper 2021/1446},
      year = {2021},
      note = {\url{https://eprint.iacr.org/2021/1446}},
      url = {https://eprint.iacr.org/2021/1446}
}
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