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On the Cost of Computing Isogenies Between Supersingular Elliptic Curves

  • Gora Adj
  • Daniel Cervantes-Vázquez
  • Jesús-Javier Chi-Domínguez
  • Alfred Menezes
  • Francisco Rodríguez-Henríquez
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11349)

Abstract

The security of the Jao-De Feo Supersingular Isogeny Diffie-Hellman (SIDH) key agreement scheme is based on the intractability of the Computational Supersingular Isogeny (CSSI) problem—computing \({\mathbb F}_{p^2}\)-rational isogenies of degrees \(2^e\) and \(3^e\) between certain supersingular elliptic curves defined over \({\mathbb F}_{p^2}\). The classical meet-in-the-middle attack on CSSI has an expected running time of \(O(p^{1/4})\), but also has \(O(p^{1/4})\) storage requirements. In this paper, we demonstrate that the van Oorschot-Wiener golden collision finding algorithm has a lower cost (but higher running time) for solving CSSI, and thus should be used instead of the meet-in-the-middle attack to assess the security of SIDH against classical attacks. The smaller parameter p brings significantly improved performance for SIDH.

Notes

Acknowledgements

We thank Steven Galbraith for the suggestion to traverse the MITM trees using depth-first search. We also thank Sam Jaques for the many discussions on Grover’s and Tani’s algorithms.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Gora Adj
    • 1
  • Daniel Cervantes-Vázquez
    • 2
  • Jesús-Javier Chi-Domínguez
    • 2
  • Alfred Menezes
    • 1
  • Francisco Rodríguez-Henríquez
    • 2
  1. 1.Department of Combinatorics and OptimizationUniversity of WaterlooWaterlooCanada
  2. 2.Computer Science DepartmentCINVESTAV-IPNMexico CityMexico

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