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Cover Attacks for Elliptic Curves over Cubic Extension Fields
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| Abstract: | We give a new approach to the elliptic curve discrete logarithm problem over cubic extension fields $${\mathbb {F}}_{q^3}$$ F q 3 . It is based on a transfer: First an $${\mathbb {F}}_q$$ F q -rational $$(\ell ,\ell ,\ell )$$ ( ℓ , ℓ , ℓ ) -isogeny from the Weil restriction of the elliptic curve under consideration with respect to $${\mathbb {F}}_{q^3}/{\mathbb {F}}_q$$ F q 3 / F q to the Jacobian variety of a genus three curve over $${\mathbb {F}}_q$$ F q is applied and then the problem is solved in the Jacobian via index-calculus attacks. Although it uses no covering maps in the construction of the desired homomorphism, this method is, in a sense, a kind of cover attack. As a result, it is possible to solve the discrete logarithm problem in some elliptic curve groups of prime order over $${\mathbb {F}}_{q^3}$$ F q 3 in a time of $${\tilde{O}}(q)$$ O ~ ( q ) . |
BibTeX
@article{jofc-2023-33319,
title={Cover Attacks for Elliptic Curves over Cubic Extension Fields},
journal={Journal of Cryptology},
publisher={Springer},
volume={36},
doi={10.1007/s00145-023-09474-2},
author={Song Tian},
year=2023
}