Classical and Quantum Algorithms for Isogeny-based Cryptography

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Date

2015-09-30

Authors

Sankar, Anirudh

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University of Waterloo

Abstract

Isogeny-based cryptography using supersingular elliptic curves --- most prominently, the constructions of De Feo-Jao-Plut --- is one of the few practical candidates for post-quantum public key cryptography. Its formidable security claim is earned through the continual exploration of quantum algorithms for `isogeny problems' and the assessment of the threat they pose to supersingular isogeny-based cryptography. We survey the rich history of classical and quantum algorithms for isogeny problems, and close with an original result --- a quantum algorithm for the general supersingular isogeny problem, based on the discovery of Delfs and Galbraith in 2013 --- that has exponential-complexity in general and subexponential complexity in an important sub-case. As yet, this algorithm poses a limited threat to the schemes of De Feo-Jao-Plut; however, it is an important algorithm to consider, for it provides insight into the structure of supersingular curves and the isogenies between them, and may lead to newer destructive quantum algorithms.

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Keywords

cryptography, public-key, elliptic curve, isogeny, post-quantum

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