Paper 2020/010

Faster point compression for elliptic curves of $j$-invariant $0$

Dmitrii Koshelev


The article provides a new double point compression method (to $2\lceil \log_2(q) \rceil + 4$ bits) for an elliptic $\mathbb{F}_{\!q}$-curve $E_b\!: y^2 = x^3 + b$ of $j$-invariant $0$ over a finite field $\mathbb{F}_{\!q}$ such that $q \equiv 1 \ (\mathrm{mod} \ 3)$. More precisely, we obtain explicit simple formulas transforming the coordinates $x_0,y_0,x_1,y_1$ of two points $P_0, P_1 \in E(\mathbb{F}_{\!q})$ to some two elements of $\mathbb{F}_{\!q}$ with four auxiliary bits. In order to recover (in the decompression stage) the points $P_0, P_1$ it is proposed to extract a sixth root $\sqrt[6]{Z} \in \mathbb{F}_{\!q}$ of some element $Z \in \mathbb{F}_{\!q}$. It is known that for $q \equiv 3 \ (\mathrm{mod} \ 4)$, $q \not\equiv 1 \ (\mathrm{mod} \ 27)$ this can be implemented by means of just one exponentiation in $\mathbb{F}_{\!q}$. Therefore the new compression method seems to be much faster than the classical one with the coordinates $x_0, x_1$, whose decompression stage requires two exponentiations in $\mathbb{F}_{\!q}$. We also successfully adapt the new approach for compressing one $\mathbb{F}_{\!q^2}$-point on a curve $E_b$ with $b \in \mathbb{F}_{\!q^2}^*$.

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Preprint. MINOR revision.
finite fieldspairing-based cryptographyelliptic curves of $j$-invariant $0$point compression
Contact author(s)
dimitri koshelev @ gmail com
2021-09-11: last of 5 revisions
2020-01-06: received
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      author = {Dmitrii Koshelev},
      title = {Faster point compression for elliptic curves of $j$-invariant $0$},
      howpublished = {Cryptology ePrint Archive, Paper 2020/010},
      year = {2020},
      note = {\url{}},
      url = {}
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