Paper 2023/1409

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

Jonas Meers, Ruhr University Bochum
Julian Nowakowski, Ruhr University Bochum

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 \({\mathsf{CDH}}\) of two curves, an adversary must compute the shared curve \(E_{AB}={\mathsf{CDH}}(E_A,E_B)\). We show that we can recover \(E_{AB}\) in polynomial time by using Coppersmith's method as long as the oracle outputs \({\frac{13}{24}} + \varepsilon \approx 54\%\) (CSIDH) and \({\frac{31}{41}} + \varepsilon \approx 76\%\) (CSURF) of the most significant bits of the \({\mathsf{CDH}}\), 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.

Available format(s)
Public-key cryptography
Publication info
Published by the IACR in ASIACRYPT 2023
CoppersmithIsogeniesCSIDHCSURFHidden Number ProblemCryptanalysis
Contact author(s)
jonas lehmann-c6j @ rub de
julian nowakowski @ rub de
2023-09-24: approved
2023-09-19: received
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      author = {Jonas Meers and Julian Nowakowski},
      title = {Solving the Hidden Number Problem for CSIDH and CSURF via Automated Coppersmith},
      howpublished = {Cryptology ePrint Archive, Paper 2023/1409},
      year = {2023},
      note = {\url{}},
      url = {}
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