A Quantum Algorithm for Computing Isogenies between Supersingular Elliptic Curves

In this paper, we describe a quantum algorithm for computing an isogeny between any two supersingular elliptic curves defined over a given finite field. The complexity of our method is in \(\tilde{O}(p^{1/4})\) where \(p\) is the characteristic of the base field. Our method is an asymptotic improvement over the previous fastest known method which had complexity \(\tilde{O}(p^{1/2})\) (on both classical and quantum computers). We also discuss the cryptographic relevance of our algorithm.

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