International Association for Cryptologic Research

International Association
for Cryptologic Research

CryptoDB

Solving the Hidden Number Problem for CSIDH and CSURF via Automated Coppersmith

Authors:
Jonas Meers , Ruhr University Bochum, Germany
Julian Nowakowski , Ruhr University Bochum, Germany
Download:
Search ePrint
Search Google
Presentation: Slides
Conference: ASIACRYPT 2023
Abstract: We define and analyze the Commutative Isogeny Hidden Number Problem which is the natural analogue of the Hidden Number Problem in the CSIDH and CSURF setting. In short, the task is as follows: Given two supersingular elliptic curves \(E_A\), \(E_B\) and access to an oracle that outputs some of the most significant bits of the \(\ensuremath{\mathsf{CDH}}\) of two curves, an adversary must compute the shared curve \(E_{AB}=\ensuremath{\mathsf{CDH}}(E_A,E_B)\). We show that we can recover \(E_{AB}\) in polynomial time by using Coppersmith's method as long as the oracle outputs \(\ensuremath{\frac{13}{24}} + \varepsilon \approx 54\%\) (CSIDH) and \(\ensuremath{\frac{31}{41}} + \varepsilon \approx 76\%\) (CSURF) of the most significant bits of the \(\ensuremath{\mathsf{CDH}}\), where $\varepsilon > 0$ is an arbitrarily small constant. To this end, we give a purely combinatorial restatement of Coppersmith's method, effectively concealing the intricate aspects of lattice theory and allowing for near-complete automation. By leveraging this approach, we attain recovery attacks with $\varepsilon$ close to zero within a few minutes of computation.
BibTeX
@inproceedings{asiacrypt-2023-33545,
  title={Solving the Hidden Number Problem for CSIDH and CSURF via Automated Coppersmith},
  publisher={Springer-Verlag},
  author={Jonas Meers and Julian Nowakowski},
  year=2023
}