Abstract
We give a new approach to the elliptic curve discrete logarithm problem over cubic extension fields \({\mathbb {F}}_{q^3}\). It is based on a transfer: First an \({\mathbb {F}}_q\)-rational \((\ell ,\ell ,\ell )\)-isogeny from the Weil restriction of the elliptic curve under consideration with respect to \({\mathbb {F}}_{q^3}/{\mathbb {F}}_q\) to the Jacobian variety of a genus three curve over \({\mathbb {F}}_q\) 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 \({\mathbb {F}}_{q^3}\) in a time of \({\tilde{O}}(q)\).
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Acknowledgements
I thank Jean-Marc Couveignes for answering questions about his paper and the anonymous reviewers for offering a number of comments and suggestions for improvements.
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Communicated by Jung Hee Cheon.
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Song Tian is supported by the National Natural Science Foundation of China under Grants Nos. 61802401, 62172412 and U1936209.
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Tian, S. Cover Attacks for Elliptic Curves over Cubic Extension Fields. J Cryptol 36, 32 (2023). https://doi.org/10.1007/s00145-023-09474-2
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DOI: https://doi.org/10.1007/s00145-023-09474-2