Papers › Explicit isogenies in quadratic time in any characteristic
Explicit isogenies in quadratic time in any characteristic
Luca De Feo, Cyril Hugounenq, Jérôme Plût, Éric Schost
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Consider two elliptic curves E,E′ defined over the finite field 𝔽_q, and suppose that there exists an isogeny ψ between E and E′. We propose an algorithm that determines ψ from the knowledge of E, E′ and of its degree r, by using the structure of the ℓ-torsion of the curves (where ℓ is a prime different from the characteristic p of the base field). Our approach is inspired by a previous algorithm due to Couveignes, that involved computations using the p-torsion on the curves. The most refined version of that algorithm, due to De Feo, has a complexity of Õ(r²) pᴼ⁽¹⁾ base field operations. On the other hand, the cost of our algorithm is Õ(r² + √(r) log(q)); this makes it an interesting alternative for the medium- and large-characteristic cases.
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