An efficient key recovery attack on SIDH

Wouter Castryck; Thomas Decru

Paper 2022/975

An efficient key recovery attack on SIDH

Wouter Castryck

, KU Leuven

Thomas Decru

, KU Leuven

Abstract

We present an efficient key recovery attack on the Supersingular Isogeny Diffie-Hellman protocol (SIDH). The attack is based on Kani's "reducibility criterion" for isogenies from products of elliptic curves and strongly relies on the torsion point images that Alice and Bob exchange during the protocol. If we assume knowledge of the endomorphism ring of the starting curve then the classical running time is polynomial in the input size (heuristically), apart from the factorization of a small number of integers that only depend on the system parameters. The attack is particularly fast and easy to implement if one of the parties uses 2-isogenies and the starting curve comes equipped with a non-scalar endomorphism of very small degree; this is the case for SIKE, the instantiation of SIDH that recently advanced to the fourth round of NIST's standardization effort for post-quantum cryptography. Our Magma implementation breaks SIKEp434, which aims at security level 1, in about ten minutes on a single core.

Note: Fixed a typo in Section 5.2.

Metadata

Available format(s): PDF
Category: Public-key cryptography
Publication info: Published by the IACR in EUROCRYPT 2023
Keywords: SIDH isogeny-based cryptography post-quantum cryptography
Contact author(s): wouter castryck @ esat kuleuven be
thomas decru @ esat kuleuven be
History: 2023-05-15: last of 4 revisions; 2022-07-30: received; See all versions
Short URL: https://ia.cr/2022/975
License: CC BY

BibTeX

@misc{cryptoeprint:2022/975,
      author = {Wouter Castryck and Thomas Decru},
      title = {An efficient key recovery attack on {SIDH}},
      howpublished = {Cryptology {ePrint} Archive, Paper 2022/975},
      year = {2022},
      url = {https://eprint.iacr.org/2022/975}
}