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Hashing to elliptic curves of j-invariant 1728

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Abstract

This article generalizes the simplified Shallue–van de Woestijne–Ulas (SWU) method of a deterministic finite field mapping \(h\!: \mathbb {F}_{q} \to E_{a}(\mathbb {F}_{q})\) to the case of any elliptic \(\mathbb {F}_{q}\)-curve Ea : y2 = x3 − ax of j-invariant 1728. In comparison with the (classical) SWU method the simplified SWU method allows to avoid one quadratic residuosity test in the field \(\mathbb {F}_{q}\), which is a quite painful operation in cryptography with regard to timing attacks. More precisely, in order to derive h we obtain a rational \(\mathbb {F}_{q}\)-curve C (and its explicit quite simple proper \(\mathbb {F}_{q}\)-parametrization) on the Kummer surface \(K^{\prime }\) associated with the direct product \({E_{a}} \times {E_{a}^{\prime }}\), where \(E_{a}^{\prime }\) is the quadratic \(\mathbb {F}_{q}\)-twist of Ea. Our approach of finding C is based on the fact that every curve Ea has a vertical \(\mathbb {F}_{q^{2}}\)-isogeny of degree 2.

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Acknowledgements

The author expresses his deep gratitude to his scientic advisor M. Tsfasman and thanks K. Loginov, K. Shramov for their help and useful comments.

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  1. Versailles Laboratory of Mathematics, Versailles Saint-Quentin-en-Yvelines University, Versailles, France

    Dmitrii Koshelev

  2. Algebra and Number Theory Laboratory, Institute for Information Transmission Problems, Moscow, Russia

    Dmitrii Koshelev

  3. Department of Discrete Mathematics, Moscow Institute of Physics and Technology, Dolgoprudny, Russia

    Dmitrii Koshelev

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Koshelev, D. Hashing to elliptic curves of j-invariant 1728. Cryptogr. Commun. 13, 479–494 (2021). https://doi.org/10.1007/s12095-021-00478-y

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