Patient Zero & Patient Six: Zero-Value and Correlation Attacks on CSIDH and SIKE

Recent works have started side-channel analysis on SIKE and show the vulnerability of isogeny-based systems to zero-value attacks. In this work, we expand on such attacks by analyzing the behavior of the zero curve \(E_0\) and six curve \(E_6\) in CSIDH and SIKE. We demonstrate an attack on static-key CSIDH and SIKE implementations that recovers bits of the secret key by observing via zero-value-based resp. exploiting correlation-collision-based side-channel analysis whether secret isogeny walks pass over the zero or six curve. We apply this attack to fully recover secret keys of SIKE and two state-of-the-art CSIDH-based implementations: CTIDH and SQALE. We show the feasibility of exploiting side-channel information for the proposed attacks based on simulations with various realistic noise levels. Additionally, we discuss countermeasures to prevent zero-value and correlation-collision attacks against CSIDH and SIKE in our attacker model.