Cover attacks for elliptic curves with prime order

12/13/2020

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by   Song Tian, et al.

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We give a new approach to the elliptic curve discrete logarithm problem over cubic extension fields 𝔽_q^3. It is based on a transfer: First an 𝔽_q-rational (ℓ,ℓ,ℓ)-isogeny from the Weil restriction of the elliptic curve under consideration with respect to 𝔽_q^3/𝔽_q to the Jacobian variety of a genus three curve over 𝔽_q is applied and then the problem is solved in the Jacobian via the index-calculus attacks. Although using 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 𝔽_q^3 in a time of Õ(q).