Faster Algorithms for Isogeny Problems Using Torsion Point Images
There is a recent trend in cryptography to construct protocols based on the hardness of computing isogenies between supersingular elliptic curves. Two prominent examples are Jao-De Feo’s key exchange protocol and the resulting encryption scheme by De Feo-Jao-Plût. One particularity of the isogeny problems underlying these protocols is that some additional information is given as input, namely the image of some torsion points with order coprime to the isogeny. This additional information was used in several active attacks against the protocols but the current best passive attacks make no use of it at all.
In this paper, we provide new algorithms that exploit the additional information provided in isogeny protocols to speed up the resolution of the underlying problems. Our techniques lead to heuristic polynomial-time key recovery on two non-standard variants of De Feo-Jao-Plût’s protocols in plausible attack models. This shows that at least some isogeny problems are easier to solve when additional information is leaked.
We thank Bryan Birch, Jonathan Bootle, Luca De Feo, Steven Galbraith, Chloe Martindale, Lorenz Panny and Yan Bo Ti, as well as the anonymous reviewers of the Asiacrypt 2017 conference for their useful comments on preliminary versions of this paper. This work was developed while the author was at the Mathematical Institute of the University of Oxford, funded by a research grant from the UK government.
- 3.Coggia, D.: Implémentation d’une variante du protocole de key-exchange SIDH (2017). https://github.com/dnlcog/sidh_variant
- 12.Kohel, D.: Endomorphism rings of elliptic curves over finite fields. PhD thesis, University of California, Berkeley (1996)Google Scholar
- 14.Petit, C.: Faster algorithms for isogeny problems using torsion point images. IACR Cryptology ePrint Archive, 2017:571 (2017)Google Scholar
- 15.Petit, C., Lauter, K.: Hard and easy problems in supersingular isogeny graphs (2017)Google Scholar
- 17.Rostovtsev, A., Stolbunov, A.: Public-key cryptosystem based on isogenies. Cryptology ePrint Archive, Report 2006/145 (2006). http://eprint.iacr.org/
- 19.Simon, D.: Quadratic equations in dimensions 4, 5 and more. Preprint (2005). http://www.math.unicaen.fr/~simon/
- 22.Fieker, C., Steel, A., Bosma, W., Cannon, J.J. (eds.): Handbook of Magma functions, edition 2.20 (2013). http://magma.maths.usyd.edu.au/magma/
- 23.Xi, S., Tian, H., Wang, Y.: Toward quantum-resistant strong designated verifier signature from isogenies. Int. J. Grid Util. Comput. 5(2), 292–296 (2012)Google Scholar
- 24.Yoo, Y., Azarderakhsh, R., Jalali, A., Jao, D., Soukharev, V.: A post-quantum digital signature scheme based on supersingular isogenies. Financial Crypto (2017)Google Scholar