An adaptive attack on 2-SIDH

Abstract

We present a polynomial-time adaptive attack on the 2-SIDH protocol. The 2-SIDH protocol is a special instance of the countermeasure proposed by Azarderakhsh, Jao and Leonardi to perform isogeny-based key exchange with static keys in the presence of an adaptive attack. This countermeasure has also been recently explicitly proposed by Kayacan. Our attack extends the adaptive attack by Galbraith, Petit, Shani and Ti (GPST) to recover a static secret key using malformed points. The extension of GPST is non-trivial and requires learning additional information. In particular, the attack needs to recover intermediate elliptic curves in the isogeny path, and points on them. We also discuss how to extend the attack to k-SIDH when k>2 and explain that the attack complexity is exponential in k.

Keywords:

• Cryptography
• elliptic curves
• key exchange
• post-Quantum
• Isogeny-based cryptography

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Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by Royal Society of New Zealand Marsden project [16-UOA-144], NSERC Michael Smith Foreign Study Supplement [533577-2018] and Ministry of Business, Innovation & Employment Catalyst project [UOAX1933].